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RD Sharma solutions for Class 10 Mathematics chapter 11 - Trigonometric Identities

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RD Sharma 10 Mathematics

10 Mathematics - Shaalaa.com

Chapter 11: Trigonometric Identities

Ex. 11.10Ex. 11.20Others

Chapter 11: Trigonometric Identities Exercise 11.10 solutions [Pages 43 - 47]

Ex. 11.10 | Q 1 | Page 43

Prove the following trigonometric identities:

`(1 - cos^2 A) cosec^2 A = 1`

Ex. 11.10 | Q 1 | Page 43

Prove the following trigonometric identities:

`(1 - cos^2 A) cosec^2 A = 1`

Ex. 11.10 | Q 2 | Page 43

Prove the following trigonometric identities

(1 + cot2 A) sin2 A = 1

Ex. 11.10 | Q 2 | Page 43

Prove the following trigonometric identities

(1 + cot2 A) sin2 A = 1

Ex. 11.10 | Q 3 | Page 43

Prove the following trigonometric identities.

tan2θ cos2θ = 1 − cos2θ

Ex. 11.10 | Q 3 | Page 43

Prove the following trigonometric identities.

tan2θ cos2θ = 1 − cos2θ

Ex. 11.10 | Q 4 | Page 43

Prove the following trigonometric identities.

`cosec theta sqrt(1 - cos^2 theta) = 1`

Ex. 11.10 | Q 4 | Page 43

Prove the following trigonometric identities.

`cosec theta sqrt(1 - cos^2 theta) = 1`

Ex. 11.10 | Q 5 | Page 43

Prove the following trigonometric identities.

(sec2 θ − 1) (cosec2 θ − 1) = 1

Ex. 11.10 | Q 5 | Page 43

Prove the following trigonometric identities.

(sec2 θ − 1) (cosec2 θ − 1) = 1

Ex. 11.10 | Q 6 | Page 43

Prove the following trigonometric identities.

`tan theta + 1/tan theta = sec theta cosec theta`

Ex. 11.10 | Q 6 | Page 43

Prove the following trigonometric identities.

`tan theta + 1/tan theta = sec theta cosec theta`

Ex. 11.10 | Q 7 | Page 43

Prove the following trigonometric identities

`cos theta/(1 - sin theta) = (1 + sin theta)/cos theta`

Ex. 11.10 | Q 7 | Page 43

Prove the following trigonometric identities

`cos theta/(1 - sin theta) = (1 + sin theta)/cos theta`

Ex. 11.10 | Q 8 | Page 43

Prove the following trigonometric identities.

`cos theta/(1 + sin theta) = (1 - sin theta)/cos theta`

Ex. 11.10 | Q 8 | Page 43

Prove the following trigonometric identities.

`cos theta/(1 + sin theta) = (1 - sin theta)/cos theta`

Ex. 11.10 | Q 9 | Page 43

Prove the following trigonometric identities

`cos^2 A + 1/(1 + cos^2 A) = 1`

Ex. 11.10 | Q 9 | Page 43

Prove the following trigonometric identities

`cos^2 A + 1/(1 + cos^2 A) = 1`

Ex. 11.10 | Q 10 | Page 43

Prove the following trigonometric identities.

`sin^2 A + 1/(1 + tan^2 A) = 1`

Ex. 11.10 | Q 10 | Page 43

Prove the following trigonometric identities.

`sin^2 A + 1/(1 + tan^2 A) = 1`

Ex. 11.10 | Q 11 | Page 43

Prove the following trigonometric identities.

`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`

Ex. 11.10 | Q 11 | Page 43

Prove the following trigonometric identities.

`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`

Ex. 11.10 | Q 12 | Page 43

Prove the following trigonometric identities.

`(1 - cos theta)/sin theta = sin theta/(1 + cos theta)`

Ex. 11.10 | Q 12 | Page 43

Prove the following trigonometric identities.

`(1 - cos theta)/sin theta = sin theta/(1 + cos theta)`

Ex. 11.10 | Q 13 | Page 44

Prove the following trigonometric identities.

`sin theta/(1 - cos theta) =  cosec theta + cot theta`

Ex. 11.10 | Q 13 | Page 44

Prove the following trigonometric identities.

`sin theta/(1 - cos theta) =  cosec theta + cot theta`

Ex. 11.10 | Q 14 | Page 44

Prove the following trigonometric identities.

`(1 - sin theta)/(1 + sin theta) = (sec theta - tan theta)^2`

Ex. 11.10 | Q 14 | Page 44

Prove the following trigonometric identities.

`(1 - sin theta)/(1 + sin theta) = (sec theta - tan theta)^2`

Ex. 11.10 | Q 15 | Page 44

Prove the following trigonometric identities.

(cosecθ + sinθ) (cosecθ − sinθ) = cot2 θ + cos2θ

Ex. 11.10 | Q 15 | Page 44

Prove the following trigonometric identities.

(cosecθ + sinθ) (cosecθ − sinθ) = cot2 θ + cos2θ

Ex. 11.10 | Q 16 | Page 44

Prove the following trigonometric identities.

`((1 + cot^2 theta) tan theta)/sec^2 theta = cot theta`

Ex. 11.10 | Q 16 | Page 44

Prove the following trigonometric identities.

`((1 + cot^2 theta) tan theta)/sec^2 theta = cot theta`

Ex. 11.10 | Q 17 | Page 44

Prove the following trigonometric identities.

(secθ + cosθ) (secθ − cosθ) = tan2θ + sin2θ

Ex. 11.10 | Q 17 | Page 44

Prove the following trigonometric identities.

(secθ + cosθ) (secθ − cosθ) = tan2θ + sin2θ

Ex. 11.10 | Q 18 | Page 44

Prove the following trigonometric identities.

secA (1 − sinA) (secA + tanA) = 1

Ex. 11.10 | Q 18 | Page 44

Prove the following trigonometric identities.

secA (1 − sinA) (secA + tanA) = 1

Ex. 11.10 | Q 19 | Page 44

Prove the following trigonometric identities.

(cosecA − sinA) (secA − cosA) (tanA + cotA) = 1

Ex. 11.10 | Q 19 | Page 44

Prove the following trigonometric identities.

(cosecA − sinA) (secA − cosA) (tanA + cotA) = 1

Ex. 11.10 | Q 20 | Page 44

Prove the following trigonometric identities.

`tan^2 theta - sin^2 theta tan^2 theta sin^2 theta`

Ex. 11.10 | Q 20 | Page 44

Prove the following trigonometric identities.

`tan^2 theta - sin^2 theta tan^2 theta sin^2 theta`

Ex. 11.10 | Q 21 | Page 44

Prove the following trigonometric identities.

(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1

Ex. 11.10 | Q 21 | Page 44

Prove the following trigonometric identities.

(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1

Ex. 11.10 | Q 22 | Page 44

Prove the following trigonometric identities.

sin2 A cot2 A + cos2 A tan2 A = 1

Ex. 11.10 | Q 22 | Page 44

Prove the following trigonometric identities.

sin2 A cot2 A + cos2 A tan2 A = 1

Ex. 11.10 | Q 23.1 | Page 44

Prove the following trigonometric identities.

`cot theta - tan theta = (2 cos^2 theta - 1)/(sin theta cos theta)`

Ex. 11.10 | Q 23.1 | Page 44

Prove the following trigonometric identities.

`cot theta - tan theta = (2 cos^2 theta - 1)/(sin theta cos theta)`

Ex. 11.10 | Q 23.2 | Page 44

Prove the following trigonometric identities.

`tan theta - cot theta = (2 sin^2 theta - 1)/(sin theta cos theta)`

Ex. 11.10 | Q 23.2 | Page 44

Prove the following trigonometric identities.

`tan theta - cot theta = (2 sin^2 theta - 1)/(sin theta cos theta)`

Ex. 11.10 | Q 24 | Page 44

Prove the following trigonometric identities.

`(cos^2 theta)/sin theta - cosec theta +  sin theta  = 0`

Ex. 11.10 | Q 24 | Page 44

Prove the following trigonometric identities.

`(cos^2 theta)/sin theta - cosec theta +  sin theta  = 0`

Ex. 11.10 | Q 25 | Page 44

Prove the following trigonometric identities.

`1/(1 + sin A) + 1/(1 - sin A) =  2sec^2 A`

Ex. 11.10 | Q 25 | Page 44

Prove the following trigonometric identities.

`1/(1 + sin A) + 1/(1 - sin A) =  2sec^2 A`

Ex. 11.10 | Q 26 | Page 44

Prove the following trigonometric identities.

`(1 + sin theta)/cos theta + cos theta/(1 + sin theta) = 2 sec theta`

Ex. 11.10 | Q 26 | Page 44

Prove the following trigonometric identities.

`(1 + sin theta)/cos theta + cos theta/(1 + sin theta) = 2 sec theta`

Ex. 11.10 | Q 27 | Page 44

Prove the following trigonometric identities

`((1 + sin theta)^2 + (1 + sin theta)^2)/(2cos^2 theta) =  (1 + sin^2 theta)/(1 - sin^2 theta)`

Ex. 11.10 | Q 27 | Page 44

Prove the following trigonometric identities

`((1 + sin theta)^2 + (1 + sin theta)^2)/(2cos^2 theta) =  (1 + sin^2 theta)/(1 - sin^2 theta)`

Ex. 11.10 | Q 28 | Page 44

Prove the following trigonometric identities

`(1 + tan^2 theta)/(1 + cot^2 theta) = ((1 - tan theta)/(1 - cot theta))^2 = tan^2 theta`

Ex. 11.10 | Q 28 | Page 44

Prove the following trigonometric identities

`(1 + tan^2 theta)/(1 + cot^2 theta) = ((1 - tan theta)/(1 - cot theta))^2 = tan^2 theta`

Ex. 11.10 | Q 29 | Page 44

Prove the following trigonometric identities.

`(1 + sec theta)/sec theta = (sin^2 theta)/(1 - cos theta)`

Ex. 11.10 | Q 29 | Page 44

Prove the following trigonometric identities.

`(1 + sec theta)/sec theta = (sin^2 theta)/(1 - cos theta)`

Ex. 11.10 | Q 30 | Page 44

Prove the following trigonometric identities.

`tan theta/(1 - cot theta) + cot theta/(1 - tan theta) = 1 + tan theta + cot theta`

Ex. 11.10 | Q 30 | Page 44

Prove the following trigonometric identities.

`tan theta/(1 - cot theta) + cot theta/(1 - tan theta) = 1 + tan theta + cot theta`

Ex. 11.10 | Q 31 | Page 44

Prove the following trigonometric identities.

sec6θ = tan6θ + 3 tan2θ sec2θ + 1

Ex. 11.10 | Q 31 | Page 44

Prove the following trigonometric identities.

sec6θ = tan6θ + 3 tan2θ sec2θ + 1

Ex. 11.10 | Q 32 | Page 44

Prove the following trigonometric identities

cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1

Ex. 11.10 | Q 32 | Page 44

Prove the following trigonometric identities

cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1

Ex. 11.10 | Q 33 | Page 44

Prove the following trigonometric identities.

`((1 + tan^2 theta)cot theta)/(cosec^2 theta)   = tan theta`

Ex. 11.10 | Q 33 | Page 44

Prove the following trigonometric identities.

`((1 + tan^2 theta)cot theta)/(cosec^2 theta)   = tan theta`

Ex. 11.10 | Q 34 | Page 44

Prove the following trigonometric identities.

`(1 + cos A)/sin^2 A = 1/(1 - cos A)`

Ex. 11.10 | Q 34 | Page 44

Prove the following trigonometric identities.

`(1 + cos A)/sin^2 A = 1/(1 - cos A)`

Ex. 11.10 | Q 35 | Page 44

Prove the following trigonometric identities.

`(sec A - tan A)/(sec A + tan A) = (cos^2 A)/(1 + sin A)^2`

Ex. 11.10 | Q 35 | Page 44

Prove the following trigonometric identities.

`(sec A - tan A)/(sec A + tan A) = (cos^2 A)/(1 + sin A)^2`

Ex. 11.10 | Q 36 | Page 44

Prove the following trigonometric identities.

`(1 + cos A)/sin A = sin A/(1 - cos A)`

Ex. 11.10 | Q 36 | Page 44

Prove the following trigonometric identities.

`(1 + cos A)/sin A = sin A/(1 - cos A)`

Ex. 11.10 | Q 37 | Page 44

Prove the following trigonometric identities.

`sqrt((1 + sin A)/(1 - sin A)) = sec A + tan A`

Ex. 11.10 | Q 37 | Page 44

Prove the following trigonometric identities.

`sqrt((1 + sin A)/(1 - sin A)) = sec A + tan A`

Ex. 11.10 | Q 38 | Page 44

Prove the following trigonometric identities.

`sqrt((1 - cos A)/(1 + cos A)) = cosec A - cot A`

Ex. 11.10 | Q 38 | Page 44

Prove the following trigonometric identities.

`sqrt((1 - cos A)/(1 + cos A)) = cosec A - cot A`

Ex. 11.10 | Q 39 | Page 45

`Prove the following trigonometric identities.

`(sec A - tan A)^2 = (1 - sin A)/(1 +  sin A)`

Ex. 11.10 | Q 39 | Page 45

`Prove the following trigonometric identities.

`(sec A - tan A)^2 = (1 - sin A)/(1 +  sin A)`

Ex. 11.10 | Q 40 | Page 45

Prove the following trigonometric identities.

`(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`

Ex. 11.10 | Q 40 | Page 45

Prove the following trigonometric identities.

`(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`

Ex. 11.10 | Q 41 | Page 45

Prove the following trigonometric identities.

`1/(sec A - 1) + 1/(sec A + 1) = 2 cosec A cot A`

Ex. 11.10 | Q 41 | Page 45

Prove the following trigonometric identities.

`1/(sec A - 1) + 1/(sec A + 1) = 2 cosec A cot A`

Ex. 11.10 | Q 42 | Page 45

Prove the following trigonometric identities.

`cos A/(1 - tan A) + sin A/(1 - cot A)  = sin A + cos A`

Ex. 11.10 | Q 42 | Page 45

Prove the following trigonometric identities.

`cos A/(1 - tan A) + sin A/(1 - cot A)  = sin A + cos A`

Ex. 11.10 | Q 43 | Page 45

Prove the following trigonometric identities.

`(cosec A)/(cosec A  - 1) + (cosec A)/(cosec A = 1) = 2 sec^2 A`

Ex. 11.10 | Q 43 | Page 45

Prove the following trigonometric identities.

`(cosec A)/(cosec A  - 1) + (cosec A)/(cosec A = 1) = 2 sec^2 A`

Ex. 11.10 | Q 44 | Page 45

Prove the following trigonometric identities.

`(1 + tan^2 A) + (1 + 1/tan^2 A) = 1/(sin^2 A - sin^4 A)`

Ex. 11.10 | Q 44 | Page 45

Prove the following trigonometric identities.

`(1 + tan^2 A) + (1 + 1/tan^2 A) = 1/(sin^2 A - sin^4 A)`

Ex. 11.10 | Q 45 | Page 45

Prove the following trigonometric identities.

`(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`

Ex. 11.10 | Q 45 | Page 45

Prove the following trigonometric identities.

`(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`

Ex. 11.10 | Q 46 | Page 45

Prove the following trigonometric identities.

`(cot A - cos A)/(cot A + cos A) = (cosec A - 1)/(cosec A + 1)`

Ex. 11.10 | Q 46 | Page 45

Prove the following trigonometric identities.

`(cot A - cos A)/(cot A + cos A) = (cosec A - 1)/(cosec A + 1)`

Ex. 11.10 | Q 47.1 | Page 45

Prove the following trigonometric identities.

`(1 + cos theta + sin theta)/(1 + cos theta - sin theta) = (1 + sin theta)/cos theta`

Ex. 11.10 | Q 47.1 | Page 45

Prove the following trigonometric identities.

`(1 + cos theta + sin theta)/(1 + cos theta - sin theta) = (1 + sin theta)/cos theta`

Ex. 11.10 | Q 47.2 | Page 45

Prove the following trigonometric identities.

`(sin theta - cos theta + 1)/(sin theta + cos theta - 1) = 1/(sec theta - tan theta)`

Ex. 11.10 | Q 47.2 | Page 45

Prove the following trigonometric identities.

`(sin theta - cos theta + 1)/(sin theta + cos theta - 1) = 1/(sec theta - tan theta)`

Ex. 11.10 | Q 47.3 | Page 45

Prove the following trigonometric identities.

`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta  + cot theta`

Ex. 11.10 | Q 47.3 | Page 45

Prove the following trigonometric identities.

`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta  + cot theta`

Ex. 11.10 | Q 48 | Page 45

Prove the following trigonometric identities.

`1/(sec A + tan A) - 1/cos A = 1/cos A - 1/(sec A - tan A)`

Ex. 11.10 | Q 48 | Page 45

Prove the following trigonometric identities.

`1/(sec A + tan A) - 1/cos A = 1/cos A - 1/(sec A - tan A)`

Ex. 11.10 | Q 49 | Page 45

Prove the following trigonometric identities

tan2 A + cot2 A = sec2 A cosec2 A − 2

Ex. 11.10 | Q 49 | Page 45

Prove the following trigonometric identities

tan2 A + cot2 A = sec2 A cosec2 A − 2

Ex. 11.10 | Q 50 | Page 45

Prove the following trigonometric identities.

`(1 - tan^2 A)/(cot^2 A -1) = tan^2 A`

Ex. 11.10 | Q 50 | Page 45

Prove the following trigonometric identities.

`(1 - tan^2 A)/(cot^2 A -1) = tan^2 A`

Ex. 11.10 | Q 51 | Page 45

Prove the following trigonometric identities.

`1 + cot^2 theta/(1 + cosec theta) = cosec theta`

Ex. 11.10 | Q 51 | Page 45

Prove the following trigonometric identities.

`1 + cot^2 theta/(1 + cosec theta) = cosec theta`

Ex. 11.10 | Q 51 | Page 45

Prove the following trigonometric identities.

`1 + cot^2 theta/(1 + cosec theta) = cosec theta`

Ex. 11.10 | Q 51 | Page 45

Prove the following trigonometric identities.

`1 + cot^2 theta/(1 + cosec theta) = cosec theta`

Ex. 11.10 | Q 52 | Page 45

Prove the following trigonometric identities.

`(cos theta)/(cosec theta + 1) + (cos theta)/(cosec theta - 1) = 2 tan theta`

Ex. 11.10 | Q 52 | Page 45

Prove the following trigonometric identities.

`(cos theta)/(cosec theta + 1) + (cos theta)/(cosec theta - 1) = 2 tan theta`

Ex. 11.10 | Q 53 | Page 45

Prove the following trigonometric identities.

`(1 + cos theta - sin^2 theta)/(sin theta (1 + cos theta)) = cot theta`

Ex. 11.10 | Q 53 | Page 45

Prove the following trigonometric identities.

`(1 + cos theta - sin^2 theta)/(sin theta (1 + cos theta)) = cot theta`

Ex. 11.10 | Q 54 | Page 45

Prove the following trigonometric identities.

`(tan^3 theta)/(1 + tan^2 theta) + (cot^3 theta)/(1 + cot^2 theta) = sec theta cosec theta - 2 sin theta cos theta`

Ex. 11.10 | Q 54 | Page 45

Prove the following trigonometric identities.

`(tan^3 theta)/(1 + tan^2 theta) + (cot^3 theta)/(1 + cot^2 theta) = sec theta cosec theta - 2 sin theta cos theta`

Ex. 11.10 | Q 55 | Page 45

Prove the following trigonometric identities.

if `T_n = sin^n theta + cos^n theta`, prove that `(T_3 - T_5)/T_1 = (T_5 - T_7)/T_3`

Ex. 11.10 | Q 55 | Page 45

Prove the following trigonometric identities.

if `T_n = sin^n theta + cos^n theta`, prove that `(T_3 - T_5)/T_1 = (T_5 - T_7)/T_3`

Ex. 11.10 | Q 56 | Page 45

Prove the following trigonometric identities.

`[tan theta + 1/cos theta]^2 + [tan theta - 1/cos theta]^2 = 2((1 + sin^2 theta)/(1 - sin^2 theta))`

Ex. 11.10 | Q 56 | Page 45

Prove the following trigonometric identities.

`[tan theta + 1/cos theta]^2 + [tan theta - 1/cos theta]^2 = 2((1 + sin^2 theta)/(1 - sin^2 theta))`

Ex. 11.10 | Q 57 | Page 45

Prove the following trigonometric identities.

`(1/(sec^2 theta - cos theta) + 1/(cosec^2 theta - sin^2 theta)) sin^2 theta cos^2 theta = (1 - sin^2 theta cos^2 theta)/(2 + sin^2 theta + cos^2 theta)`

Ex. 11.10 | Q 57 | Page 45

Prove the following trigonometric identities.

`(1/(sec^2 theta - cos theta) + 1/(cosec^2 theta - sin^2 theta)) sin^2 theta cos^2 theta = (1 - sin^2 theta cos^2 theta)/(2 + sin^2 theta + cos^2 theta)`

Ex. 11.10 | Q 58 | Page 45

Prove the following trigonometric identities.

`((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (1 - cos theta)/(1 + cos theta)`

Ex. 11.10 | Q 58 | Page 45

Prove the following trigonometric identities.

`((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (1 - cos theta)/(1 + cos theta)`

Ex. 11.10 | Q 59 | Page 45

Prove the following trigonometric identities.

(sec A + tan A − 1) (sec A − tan A + 1) = 2 tan A

Ex. 11.10 | Q 59 | Page 45

Prove the following trigonometric identities.

(sec A + tan A − 1) (sec A − tan A + 1) = 2 tan A

Ex. 11.10 | Q 60 | Page 46

Prove the following trigonometric identities.

(1 + cot A − cosec A) (1 + tan A + sec A) = 2

Ex. 11.10 | Q 60 | Page 46

Prove the following trigonometric identities.

(1 + cot A − cosec A) (1 + tan A + sec A) = 2

Ex. 11.10 | Q 61 | Page 46

Prove the following trigonometric identities.

(cosec θ − sec θ) (cot θ − tan θ) = (cosec θ + sec θ) ( sec θ cosec θ − 2)

Ex. 11.10 | Q 61 | Page 46

Prove the following trigonometric identities.

(cosec θ − sec θ) (cot θ − tan θ) = (cosec θ + sec θ) ( sec θ cosec θ − 2)

Ex. 11.10 | Q 62 | Page 46

Prove the following trigonometric identities.

(sec A − cosec A) (1 + tan A + cot A) = tan A sec A − cot A cosec A

Ex. 11.10 | Q 62 | Page 46

Prove the following trigonometric identities.

(sec A − cosec A) (1 + tan A + cot A) = tan A sec A − cot A cosec A

Ex. 11.10 | Q 63 | Page 46

Prove the following trigonometric identities.

`(cos A cosec A - sin A sec A)/(cos A + sin A) = cosec A - sec A`

Ex. 11.10 | Q 63 | Page 46

Prove the following trigonometric identities.

`(cos A cosec A - sin A sec A)/(cos A + sin A) = cosec A - sec A`

Ex. 11.10 | Q 64 | Page 46

Prove the following trigonometric identities.

`sin A/(sec A + tan A - 1) + cos A/(cosec A + cot A + 1) = 1`

Ex. 11.10 | Q 64 | Page 46

Prove the following trigonometric identities.

`sin A/(sec A + tan A - 1) + cos A/(cosec A + cot A + 1) = 1`

Ex. 11.10 | Q 65 | Page 46

Prove the following trigonometric identities.

`tan A/(1 + tan^2  A)^2 + cot A/((1 + cot^2 A)) = sin A  cos A`

Ex. 11.10 | Q 65 | Page 46

Prove the following trigonometric identities.

`tan A/(1 + tan^2  A)^2 + cot A/((1 + cot^2 A)) = sin A  cos A`

Ex. 11.10 | Q 66 | Page 46

Prove the following trigonometric identities

sec4 A(1 − sin4 A) − 2 tan2 A = 1

Ex. 11.10 | Q 66 | Page 46

Prove the following trigonometric identities

sec4 A(1 − sin4 A) − 2 tan2 A = 1

Ex. 11.10 | Q 67 | Page 46

Prove the following trigonometric identities.

`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`

Ex. 11.10 | Q 67 | Page 46

Prove the following trigonometric identities.

`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`

Ex. 11.10 | Q 68 | Page 46

Prove the following trigonometric identities.

`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`

Ex. 11.10 | Q 68 | Page 46

Prove the following trigonometric identities.

`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`

Ex. 11.10 | Q 69 | Page 46

Prove the following trigonometric identities.

sin2 A cos2 B − cos2 A sin2 B = sin2 A − sin2 B

Ex. 11.10 | Q 69 | Page 46

Prove the following trigonometric identities.

sin2 A cos2 B − cos2 A sin2 B = sin2 A − sin2 B

Ex. 11.10 | Q 70 | Page 46

Prove the following trigonometric identities.

`(cot A + tan B)/(cot B + tan A) = cot A tan B`

Ex. 11.10 | Q 70 | Page 46

Prove the following trigonometric identities.

`(cot A + tan B)/(cot B + tan A) = cot A tan B`

Ex. 11.10 | Q 71 | Page 46

Prove the following trigonometric identities.

`(tan A + tan B)/(cot A + cot B) = tan A tan B`

Ex. 11.10 | Q 71 | Page 46

Prove the following trigonometric identities.

`(tan A + tan B)/(cot A + cot B) = tan A tan B`

Ex. 11.10 | Q 72 | Page 46

Prove the following trigonometric identities.

`cot^2 A cosec^2B - cot^2 B cosec^2 A = cot^2 A - cot^2 B`

Ex. 11.10 | Q 72 | Page 46

Prove the following trigonometric identities.

`cot^2 A cosec^2B - cot^2 B cosec^2 A = cot^2 A - cot^2 B`

Ex. 11.10 | Q 73 | Page 46

Prove the following trigonometric identities.

tan2 A sec2 B − sec2 A tan2 B = tan2 A − tan2 B

Ex. 11.10 | Q 73 | Page 46

Prove the following trigonometric identities.

tan2 A sec2 B − sec2 A tan2 B = tan2 A − tan2 B

Ex. 11.10 | Q 74 | Page 46

Prove the following trigonometric identities

If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that x2 − y2 = a2 − b2

Ex. 11.10 | Q 74 | Page 46

Prove the following trigonometric identities

If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that x2 − y2 = a2 − b2

Ex. 11.10 | Q 75 | Page 46

if `x/a cos theta + y/b sin theta = 1` and `x/a sin theta - y/b cos theta = 1` prove that `x^2/a^2 + y^2/b^2  = 2`

Ex. 11.10 | Q 75 | Page 46

if `x/a cos theta + y/b sin theta = 1` and `x/a sin theta - y/b cos theta = 1` prove that `x^2/a^2 + y^2/b^2  = 2`

Ex. 11.10 | Q 76 | Page 46

if `cosec theta - sin theta = a^3`, `sec theta - cos theta = b^3` prove that `a^2 b^2 (a^2 + b^2) = 1`

Ex. 11.10 | Q 76 | Page 46

if `cosec theta - sin theta = a^3`, `sec theta - cos theta = b^3` prove that `a^2 b^2 (a^2 + b^2) = 1`

Ex. 11.10 | Q 77 | Page 46

if `a cos^3 theta + 3a cos theta sin^2 theta = m, a sin^3 theta + 3 a cos^2 theta sin theta = n`Prove that `(m + n)^(2/3) + (m - n)^(2/3)`

Ex. 11.10 | Q 77 | Page 46

if `a cos^3 theta + 3a cos theta sin^2 theta = m, a sin^3 theta + 3 a cos^2 theta sin theta = n`Prove that `(m + n)^(2/3) + (m - n)^(2/3)`

Ex. 11.10 | Q 78 | Page 47

Prove the following trigonometric identities.

if x = a cos^3 theta, y = b sin^3 theta` " prove that " `(x/a)^(2/3) + (y/b)^(2/3) = 1`

Ex. 11.10 | Q 78 | Page 47

Prove the following trigonometric identities.

if x = a cos^3 theta, y = b sin^3 theta` " prove that " `(x/a)^(2/3) + (y/b)^(2/3) = 1`

Ex. 11.10 | Q 79 | Page 47

If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.

Ex. 11.10 | Q 79 | Page 47

If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.

Ex. 11.10 | Q 80 | Page 47

If a cos θ + b sin θ = m and a sin θ – b cos θ = n, prove that a2 + b2 = m2 + n2

Ex. 11.10 | Q 80 | Page 47

If a cos θ + b sin θ = m and a sin θ – b cos θ = n, prove that a2 + b2 = m2 + n2

Ex. 11.10 | Q 81 | Page 47

If cos θ + cot θ = m and cosec θ – cot θ = n, prove that mn = 1

Ex. 11.10 | Q 81 | Page 47

If cos θ + cot θ = m and cosec θ – cot θ = n, prove that mn = 1

Ex. 11.10 | Q 82 | Page 47

Prove the following trigonometric identities.

if cos A + cos2 A = 1, prove that sin2 A + sin4 A = 1

Ex. 11.10 | Q 82 | Page 47

Prove the following trigonometric identities.

if cos A + cos2 A = 1, prove that sin2 A + sin4 A = 1

Ex. 11.10 | Q 83.1 | Page 47

Prove that: `sqrt((sec theta - 1)/(sec theta + 1)) + sqrt((sec theta + 1)/(sec theta - 1)) = 2 cosec theta`

Ex. 11.10 | Q 83.1 | Page 47

Prove that: `sqrt((sec theta - 1)/(sec theta + 1)) + sqrt((sec theta + 1)/(sec theta - 1)) = 2 cosec theta`

Ex. 11.10 | Q 83.2 | Page 47

Prove that `sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta)) = 2 sec theta`

Ex. 11.10 | Q 83.2 | Page 47

Prove that `sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta)) = 2 sec theta`

Ex. 11.10 | Q 83.3 | Page 47

Prove that `sqrt((1 + cos theta)/(1 - cos theta)) + sqrt((1 - cos theta)/(1 + cos theta)) = 2 cosec theta`

Ex. 11.10 | Q 83.3 | Page 47

Prove that `sqrt((1 + cos theta)/(1 - cos theta)) + sqrt((1 - cos theta)/(1 + cos theta)) = 2 cosec theta`

Ex. 11.10 | Q 83.4 | Page 47

Prove that `(sec theta - 1)/(sec theta + 1) = ((sin theta)/(1 + cos theta))^2`

Ex. 11.10 | Q 83.4 | Page 47

Prove that `(sec theta - 1)/(sec theta + 1) = ((sin theta)/(1 + cos theta))^2`

Ex. 11.10 | Q 84 | Page 47

If cos θ + cos2 θ = 1, prove that sin12 θ + 3 sin10 θ + 3 sin8 θ + sin6 θ + 2 sin4 θ + 2 sin2 θ − 2 = 1

Ex. 11.10 | Q 84 | Page 47

If cos θ + cos2 θ = 1, prove that sin12 θ + 3 sin10 θ + 3 sin8 θ + sin6 θ + 2 sin4 θ + 2 sin2 θ − 2 = 1

Ex. 11.10 | Q 85 | Page 47

Given that:
(1 + cos α) (1 + cos β) (1 + cos γ) = (1 − cos α) (1 − cos α) (1 − cos β) (1 − cos γ)

Show that one of the values of each member of this equality is sin α sin β sin γ

Ex. 11.10 | Q 85 | Page 47

Given that:
(1 + cos α) (1 + cos β) (1 + cos γ) = (1 − cos α) (1 − cos α) (1 − cos β) (1 − cos γ)

Show that one of the values of each member of this equality is sin α sin β sin γ

Ex. 11.10 | Q 86 | Page 47

If sin θ + cos θ = x, prove that  `sin^6 theta + cos^6 theta = (4- 3(x^2 - 1)^2)/4`

Ex. 11.10 | Q 86 | Page 47

If sin θ + cos θ = x, prove that  `sin^6 theta + cos^6 theta = (4- 3(x^2 - 1)^2)/4`

Ex. 11.10 | Q 87 | Page 47

If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z c tan θ, show that `x^2/a^2 + y^2/b^2 - x^2/c^2 = 1`

Ex. 11.10 | Q 87 | Page 47

If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z c tan θ, show that `x^2/a^2 + y^2/b^2 - x^2/c^2 = 1`

Chapter 11: Trigonometric Identities Exercise 11.20 solutions [Page 54]

Ex. 11.20 | Q 1 | Page 54

if `cos theta = 4/5` find all other trigonometric ratios of angles θ

Ex. 11.20 | Q 1 | Page 54

if `cos theta = 4/5` find all other trigonometric ratios of angles θ

Ex. 11.20 | Q 2 | Page 54

if `sin theta = 1/sqrt2`  find all other trigonometric ratios of angle θ.

Ex. 11.20 | Q 2 | Page 54

if `sin theta = 1/sqrt2`  find all other trigonometric ratios of angle θ.

Ex. 11.20 | Q 3 | Page 54

if `tan theta = 1/sqrt2` find the value of `(cosec^2 theta - sec^2 theta)/(cosec^2 theta + cot^2 theta)`

Ex. 11.20 | Q 3 | Page 54

if `tan theta = 1/sqrt2` find the value of `(cosec^2 theta - sec^2 theta)/(cosec^2 theta + cot^2 theta)`

Ex. 11.20 | Q 4 | Page 54

if `tan theta = 3/4`, find the value of `(1 - cos theta)/(1 +cos theta)`

Ex. 11.20 | Q 4 | Page 54

if `tan theta = 3/4`, find the value of `(1 - cos theta)/(1 +cos theta)`

Ex. 11.20 | Q 5 | Page 54

if `tan theta = 12/5` find the value of `(1 + sin theta)/(1 -sin theta)` 

Ex. 11.20 | Q 5 | Page 54

if `tan theta = 12/5` find the value of `(1 + sin theta)/(1 -sin theta)` 

Ex. 11.20 | Q 6 | Page 54

if `cot theta = 1/sqrt3` find the value of `(1 - cos^2 theta)/(2 - sin^2 theta)`

Ex. 11.20 | Q 6 | Page 54

if `cot theta = 1/sqrt3` find the value of `(1 - cos^2 theta)/(2 - sin^2 theta)`

Ex. 11.20 | Q 7 | Page 54

if `cosec A = sqrt2` find the value of `(2 sin^2 A + 3 cot^2 A)/(4(tan^2 A - cos^2 A))`

Ex. 11.20 | Q 7 | Page 54

if `cosec A = sqrt2` find the value of `(2 sin^2 A + 3 cot^2 A)/(4(tan^2 A - cos^2 A))`

Ex. 11.20 | Q 8 | Page 54

if `cot theta = sqrt3` find the value of `(cosec^2 theta + cot^2 theta)/(cosec^2 theta - sec^2 theta)`

Ex. 11.20 | Q 8 | Page 54

if `cot theta = sqrt3` find the value of `(cosec^2 theta + cot^2 theta)/(cosec^2 theta - sec^2 theta)`

Ex. 11.20 | Q 9 | Page 54

if `3 cos theta = 1`, find the value of `(6 sin^2 theta + tan^2 theta)/(4 cos theta)`

Ex. 11.20 | Q 9 | Page 54

if `3 cos theta = 1`, find the value of `(6 sin^2 theta + tan^2 theta)/(4 cos theta)`

Ex. 11.20 | Q 10 | Page 54

if `sqrt3 tan theta = 3 sin theta` find the value of `sin^2 theta - cos^2 theta`

Ex. 11.20 | Q 10 | Page 54

if `sqrt3 tan theta = 3 sin theta` find the value of `sin^2 theta - cos^2 theta`

Chapter 11: Trigonometric Identities solutions [Pages 55 - 56]

Q 1 | Page 55

Define an identity.

Q 1 | Page 55

Define an identity.

Q 2 | Page 55

What is the value of (1 − cos2 θ) cosec2 θ? 

Q 2 | Page 55

What is the value of (1 − cos2 θ) cosec2 θ? 

Q 3 | Page 55

What is the value of (1 + cot2 θ) sin2 θ?

Q 3 | Page 55

What is the value of (1 + cot2 θ) sin2 θ?

Q 4 | Page 55

What is the value of \[\sin^2 \theta + \frac{1}{1 + \tan^2 \theta}\]

Q 4 | Page 55

What is the value of \[\sin^2 \theta + \frac{1}{1 + \tan^2 \theta}\]

Q 5 | Page 55

If sec θ + tan θ = x, write the value of sec θ − tan θ in terms of x.

Q 5 | Page 55

If sec θ + tan θ = x, write the value of sec θ − tan θ in terms of x.

Q 6 | Page 55

If cosec θ − cot θ = α, write the value of cosec θ + cot α.

Q 6 | Page 55

If cosec θ − cot θ = α, write the value of cosec θ + cot α.

Q 7 | Page 55

Write the value of cosec2 (90° − θ) − tan2 θ. 

Q 7 | Page 55

Write the value of cosec2 (90° − θ) − tan2 θ. 

Q 8 | Page 55

Write the value of sin A cos (90° − A) + cos A sin (90° − A).

Q 8 | Page 55

Write the value of sin A cos (90° − A) + cos A sin (90° − A).

Q 9 | Page 55

Write the value of \[\cot^2 \theta - \frac{1}{\sin^2 \theta}\] 

Q 9 | Page 55

Write the value of \[\cot^2 \theta - \frac{1}{\sin^2 \theta}\] 

Q 10 | Page 55

If x = a sin θ and y = b cos θ, what is the value of b2x2 + a2y2?

Q 10 | Page 55

If x = a sin θ and y = b cos θ, what is the value of b2x2 + a2y2?

Q 11 | Page 55

If \[\sin \theta = \frac{4}{5}\] what is the value of cotθ + cosecθ? 

Q 11 | Page 55

If \[\sin \theta = \frac{4}{5}\] what is the value of cotθ + cosecθ? 

Q 12 | Page 55

What is the value of 9cot2 θ − 9cosec2 θ? 

Q 12 | Page 55

What is the value of 9cot2 θ − 9cosec2 θ? 

Q 13 | Page 55

What is the value of \[6 \tan^2 \theta - \frac{6}{\cos^2 \theta}\]

 

Q 13 | Page 55

What is the value of \[6 \tan^2 \theta - \frac{6}{\cos^2 \theta}\]

 

Q 14 | Page 55

What is the value of \[\frac{\tan^2 \theta - \sec^2 \theta}{\cot^2 \theta - {cosec}^2 \theta}\]

Q 14 | Page 55

What is the value of \[\frac{\tan^2 \theta - \sec^2 \theta}{\cot^2 \theta - {cosec}^2 \theta}\]

Q 15 | Page 55

What is the value of (1 + tan2 θ) (1 − sin θ) (1 + sin θ)?

Q 15 | Page 55

What is the value of (1 + tan2 θ) (1 − sin θ) (1 + sin θ)?

Q 16 | Page 55

If \[\cos A = \frac{7}{25}\]  find the value of tan A + cot A. 

Q 16 | Page 55

If \[\cos A = \frac{7}{25}\]  find the value of tan A + cot A. 

Q 17 | Page 55

If \[\sin \theta = \frac{1}{3}\] then find the value of 2cot2 θ + 2. 

Q 17 | Page 55

If \[\sin \theta = \frac{1}{3}\] then find the value of 2cot2 θ + 2. 

Q 18 | Page 55

If \[\sin \theta = \frac{1}{3}\] then find the value of 9tan2 θ + 9. 

Q 18 | Page 55

If \[\sin \theta = \frac{1}{3}\] then find the value of 9tan2 θ + 9. 

Q 19 | Page 55

If sec2 θ (1 + sin θ) (1 − sin θ) = k, then find the value of k.

Q 19 | Page 55

If sec2 θ (1 + sin θ) (1 − sin θ) = k, then find the value of k.

Q 20 | Page 55

If cosec2 θ (1 + cos θ) (1 − cos θ) = λ, then find the value of λ. 

Q 20 | Page 55

If cosec2 θ (1 + cos θ) (1 − cos θ) = λ, then find the value of λ. 

Q 21 | Page 55

If sin2 θ cos2 θ (1 + tan2 θ) (1 + cot2 θ) = λ, then find the value of λ. 

Q 21 | Page 55

If sin2 θ cos2 θ (1 + tan2 θ) (1 + cot2 θ) = λ, then find the value of λ. 

Q 22 | Page 55

If 5x = sec θ and \[\frac{5}{x} = \tan \theta\]find the value of \[5\left( x^2 - \frac{1}{x^2} \right)\] 

Q 22 | Page 55

If 5x = sec θ and \[\frac{5}{x} = \tan \theta\]find the value of \[5\left( x^2 - \frac{1}{x^2} \right)\] 

Q 23 | Page 55

If cosec θ = 2x and \[5\left( x^2 - \frac{1}{x^2} \right)\] \[2\left( x^2 - \frac{1}{x^2} \right)\] 

Q 23 | Page 55

If cosec θ = 2x and \[5\left( x^2 - \frac{1}{x^2} \right)\] \[2\left( x^2 - \frac{1}{x^2} \right)\] 

Q 24.1 | Page 56

 Write True' or False' and justify your answer  the following : 

The value of  \[\sin \theta\] is \[x + \frac{1}{x}\] where 'x'  is a positive real number . 

Q 24.1 | Page 56

 Write True' or False' and justify your answer  the following : 

The value of  \[\sin \theta\] is \[x + \frac{1}{x}\] where 'x'  is a positive real number . 

Q 24.2 | Page 56

Write True' or False' and justify your answer the following : 

\[ \cos \theta = \frac{a^2 + b^2}{2ab}\]where and b are two distinct numbers such that ab > 0.

Q 24.2 | Page 56

Write True' or False' and justify your answer the following : 

\[ \cos \theta = \frac{a^2 + b^2}{2ab}\]where and b are two distinct numbers such that ab > 0.

Q 24.3 | Page 56

 Write True' or False' and justify your answer  the following : 

The value of  \[\cos^2 23 - \sin^2 67\]  is positive . 

Q 24.3 | Page 56

 Write True' or False' and justify your answer  the following : 

The value of  \[\cos^2 23 - \sin^2 67\]  is positive . 

Q 24.4 | Page 56

 Write True' or False' and justify your answer the following :

The value of the expression \[\sin {80}^° - \cos {80}^°\] 

Q 24.4 | Page 56

 Write True' or False' and justify your answer the following :

The value of the expression \[\sin {80}^° - \cos {80}^°\] 

Q 24.5 | Page 56

 Write True' or False' and justify your answer  the following : 

The value of sin θ+cos θ is always greater than 1 .

Q 24.5 | Page 56

 Write True' or False' and justify your answer  the following : 

The value of sin θ+cos θ is always greater than 1 .

Chapter 11: Trigonometric Identities solutions [Pages 56 - 59]

Q 1 | Page 56

If sec θ + tan θ = x, then sec θ =

  • \[\frac{x^2 + 1}{x}\]

  • \[\frac{x^2 + 1}{2x}\]

  • \[\frac{x^2 - 1}{2x}\]

  • \[\frac{x^2 - 1}{x}\]

Q 1 | Page 56

If sec θ + tan θ = x, then sec θ =

  • \[\frac{x^2 + 1}{x}\]

  • \[\frac{x^2 + 1}{2x}\]

  • \[\frac{x^2 - 1}{2x}\]

  • \[\frac{x^2 - 1}{x}\]

Q 2 | Page 56

If \[sec\theta + tan\theta = x\] then \[tan\theta =\] 

  • \[\frac{x^2 + 1}{x}\]

  • \[\frac{x^2 - 1}{x}\]

  • \[\frac{x^2 + 1}{2x}\]

  • \[\frac{x^2 - 1}{2x}\] 

Q 2 | Page 56

If \[sec\theta + tan\theta = x\] then \[tan\theta =\] 

  • \[\frac{x^2 + 1}{x}\]

  • \[\frac{x^2 - 1}{x}\]

  • \[\frac{x^2 + 1}{2x}\]

  • \[\frac{x^2 - 1}{2x}\] 

Q 3 | Page 56

\[\frac{x^2 - 1}{2x}\] is equal to 

  •  sec θ + tan θ

  •  sec θ − tan θ

  •  sec2 θ + tan2 θ

  • sec2 θ − tan2 θ

Q 3 | Page 56

\[\frac{x^2 - 1}{2x}\] is equal to 

  •  sec θ + tan θ

  •  sec θ − tan θ

  •  sec2 θ + tan2 θ

  • sec2 θ − tan2 θ

Q 4 | Page 56

The value of \[\sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}}\]

  •  cot θ − cosec θ

  •  cosec θ + cot θ

  • cosec2 θ + cot2 θ

  •  (cot θ + cosec θ)2

Q 4 | Page 56

The value of \[\sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}}\]

  •  cot θ − cosec θ

  •  cosec θ + cot θ

  • cosec2 θ + cot2 θ

  •  (cot θ + cosec θ)2

Q 5 | Page 56

sec4 A − sec2 A is equal to

  • tan2 A − tan4 A

  • tan4 A − tan2 A

  • tan4 A + tan2 A

  •  tan2 A + tan4 A

Q 5 | Page 56

sec4 A − sec2 A is equal to

  • tan2 A − tan4 A

  • tan4 A − tan2 A

  • tan4 A + tan2 A

  •  tan2 A + tan4 A

Q 6 | Page 56

cos4 A − sin4 A is equal to

  • 2 cos2 A + 1

  •  2 cos2 A − 1

  • 2 sin2 A − 1

  •  2 sin2 A + 1

Q 6 | Page 56

cos4 A − sin4 A is equal to

  • 2 cos2 A + 1

  •  2 cos2 A − 1

  • 2 sin2 A − 1

  •  2 sin2 A + 1

Q 7 | Page 57

\[\frac{\sin \theta}{1 + \cos \theta}\]is equal to 

  • \[\frac{\sin \theta}{1 + \cos \theta}\]

  • \[\frac{1 - \cos \theta}{\cos \theta}\]

  • \[\frac{1 - \cos \theta}{\cos \theta}\]

  • \[\frac{1 - \sin \theta}{\cos \theta}\]

Q 7 | Page 57

\[\frac{\sin \theta}{1 + \cos \theta}\]is equal to 

  • \[\frac{\sin \theta}{1 + \cos \theta}\]

  • \[\frac{1 - \cos \theta}{\cos \theta}\]

  • \[\frac{1 - \cos \theta}{\cos \theta}\]

  • \[\frac{1 - \sin \theta}{\cos \theta}\]

Q 8 | Page 57

\[\frac{1 - \sin \theta}{\cos \theta}\] is equal to

  •  0

  • 1

  • sin θ + cos θ

  • sin θ − cos θ

Q 8 | Page 57

\[\frac{1 - \sin \theta}{\cos \theta}\] is equal to

  •  0

  • 1

  • sin θ + cos θ

  • sin θ − cos θ

Q 9 | Page 57

The value of (1 + cot θ − cosec θ) (1 + tan θ + sec θ) is 

  • 1

  • 2

  • 4

  • 0

Q 9 | Page 57

The value of (1 + cot θ − cosec θ) (1 + tan θ + sec θ) is 

  • 1

  • 2

  • 4

  • 0

Q 10 | Page 57

\[\frac{\tan \theta}{\sec \theta - 1} + \frac{\tan \theta}{\sec \theta + 1}\] is equal to 

  • 2 tan θ

  •  2 sec θ

  •  2 cosec θ

  •  2 tan θ sec θ

Q 10 | Page 57

\[\frac{\tan \theta}{\sec \theta - 1} + \frac{\tan \theta}{\sec \theta + 1}\] is equal to 

  • 2 tan θ

  •  2 sec θ

  •  2 cosec θ

  •  2 tan θ sec θ

Q 11 | Page 57

(cosec θ − sin θ) (sec θ − cos θ) (tan θ + cot θ) is equal to

  • 0

  • 1

  •  −1

  • None of these

Q 11 | Page 57

(cosec θ − sin θ) (sec θ − cos θ) (tan θ + cot θ) is equal to

  • 0

  • 1

  •  −1

  • None of these

Q 12 | Page 57

If x = a cos θ and y = b sin θ, then b2x2 + a2y2 =

  • a2 b2

  • ab

  • a4 b4

  • a2 + b2

Q 12 | Page 57

If x = a cos θ and y = b sin θ, then b2x2 + a2y2 =

  • a2 b2

  • ab

  • a4 b4

  • a2 + b2

Q 13 | Page 57

If x = a sec θ and y = b tan θ, then b2x2 − a2y2 =

  •  ab

  • a2 − b2

  •  a2 + b2

  • a2 b2

Q 13 | Page 57

If x = a sec θ and y = b tan θ, then b2x2 − a2y2 =

  •  ab

  • a2 − b2

  •  a2 + b2

  • a2 b2

Q 14 | Page 57

\[\frac{\tan \theta}{\sec \theta - 1} + \frac{\tan \theta}{\sec \theta + 1}\] is equal to 

 

 

  • 0

  • 1

  • -1

  • 2

Q 14 | Page 57

\[\frac{\tan \theta}{\sec \theta - 1} + \frac{\tan \theta}{\sec \theta + 1}\] is equal to 

 

 

  • 0

  • 1

  • -1

  • 2

Q 15 | Page 57

2 (sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) is equal to 

  •  0

  •  1

  •  −1

  • None of these

Q 15 | Page 57

2 (sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) is equal to 

  •  0

  •  1

  •  −1

  • None of these

Q 16 | Page 57

If a cos θ + b sin θ = 4 and a sin θ − b sin θ = 3, then a2 + b2

  •  7

  • 12

  • 25

  • None of these

Q 16 | Page 57

If a cos θ + b sin θ = 4 and a sin θ − b sin θ = 3, then a2 + b2

  •  7

  • 12

  • 25

  • None of these

Q 17 | Page 57

If cot θ + b cosec θ = p and b cot θ − a cosec θ = q, then p2 − q2 

  • a2 − b2

  • b2 − a2

  • a2 + b2

  •  b − a

Q 17 | Page 57

If cot θ + b cosec θ = p and b cot θ − a cosec θ = q, then p2 − q2 

  • a2 − b2

  • b2 − a2

  • a2 + b2

  •  b − a

Q 18 | Page 57

The value of sin2 29° + sin2 61° is

  • 1

  • 0

  •  2 sin2 29°

  • 2 cos2 61° 

     

Q 18 | Page 57

The value of sin2 29° + sin2 61° is

  • 1

  • 0

  •  2 sin2 29°

  • 2 cos2 61° 

     

Q 19 | Page 57

If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then 

  • \[x^2 + y^2 + z^2 = r^2\]

  • \[x^2 + y^2 - z^2 = r^2\]

  • \[x^2 - y^2 + z^2 = r^2\]

  • \[z^2 + y^2 - x^2 = r^2\] 

Q 19 | Page 57

If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then 

  • \[x^2 + y^2 + z^2 = r^2\]

  • \[x^2 + y^2 - z^2 = r^2\]

  • \[x^2 - y^2 + z^2 = r^2\]

  • \[z^2 + y^2 - x^2 = r^2\] 

Q 20 | Page 58

If sin θ + sin2 θ = 1, then cos2 θ + cos4 θ = 

  • −1

  • 1

  • None of these

Q 20 | Page 58

If sin θ + sin2 θ = 1, then cos2 θ + cos4 θ = 

  • −1

  • 1

  • None of these

Q 21 | Page 58

If a cos θ + b sin θ = m and a sin θ − b cos θ = n, then a2 + b2 =

  • m2 − n2

  • m2n2

  •  n2 − m2

  • m2 + n2

Q 21 | Page 58

If a cos θ + b sin θ = m and a sin θ − b cos θ = n, then a2 + b2 =

  • m2 − n2

  • m2n2

  •  n2 − m2

  • m2 + n2

Q 22 | Page 58

If cos A + cos2 A = 1, then sin2 A + sin4 A =

  • −1

  • 0

  • 1

  • None of these

Q 22 | Page 58

If cos A + cos2 A = 1, then sin2 A + sin4 A =

  • −1

  • 0

  • 1

  • None of these

Q 23 | Page 58

If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, then\[\frac{x^2}{a^2} + \frac{y^2}{b^2}\]

  • \[\frac{z^2}{c^2}\]

  • \[1 - \frac{z^2}{c^2}\]

  • \[\frac{z^2}{c^2} - 1\]

  • \[1 + \frac{z^2}{c^2}\]

Q 23 | Page 58

If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, then\[\frac{x^2}{a^2} + \frac{y^2}{b^2}\]

  • \[\frac{z^2}{c^2}\]

  • \[1 - \frac{z^2}{c^2}\]

  • \[\frac{z^2}{c^2} - 1\]

  • \[1 + \frac{z^2}{c^2}\]

Q 24 | Page 58

If a cos θ − b sin θ = c, then a sin θ + b cos θ =

  • \[\pm \sqrt{a^2 + b^2 + c^2}\]

  • \[\pm \sqrt{a^2 + b^2 - c^2}\]

  • \[\pm \sqrt{c^2 - a^2 - b^2}\]

  •  None of these

Q 24 | Page 58

If a cos θ − b sin θ = c, then a sin θ + b cos θ =

  • \[\pm \sqrt{a^2 + b^2 + c^2}\]

  • \[\pm \sqrt{a^2 + b^2 - c^2}\]

  • \[\pm \sqrt{c^2 - a^2 - b^2}\]

  •  None of these

Q 25 | Page 58

9 sec2 A − 9 tan2 A is equal to

  • 1

  • 9

  • 8

  • 0

Q 25 | Page 58

9 sec2 A − 9 tan2 A is equal to

  • 1

  • 9

  • 8

  • 0

Q 26 | Page 58

(1 + tan θ + sec θ) (1 + cot θ − cosec θ) =

  • 0

  • 1

  • 1

  • -1

  • none of these

Q 26 | Page 58

(1 + tan θ + sec θ) (1 + cot θ − cosec θ) =

  • 0

  • 1

  • 1

  • -1

  • none of these

Q 27 | Page 58

(sec A + tan A) (1 − sin A) =

  •  sec A

  •  sin A

  •  cosec A

  • cos A

Q 27 | Page 58

(sec A + tan A) (1 − sin A) =

  •  sec A

  •  sin A

  •  cosec A

  • cos A

Q 28 | Page 58

\[\frac{1 + \tan^2 A}{1 + \cot^2 A}\]is equal to

  •  sec2 A

  • −1

  •  cot2 A

  •  tan2 A

Q 28 | Page 58

\[\frac{1 + \tan^2 A}{1 + \cot^2 A}\]is equal to

  •  sec2 A

  • −1

  •  cot2 A

  •  tan2 A

Q 29 | Page 58

If sin \[\theta - \cos \theta = 0\]  then the value of \[\sin^4 \theta + \cos^4 \theta\]

  • 1

  • \[- 1\]

  • \[\frac{1}{2}\]

  • \[\frac{1}{4}\]

Q 29 | Page 58

If sin \[\theta - \cos \theta = 0\]  then the value of \[\sin^4 \theta + \cos^4 \theta\]

  • 1

  • \[- 1\]

  • \[\frac{1}{2}\]

  • \[\frac{1}{4}\]

Q 30 | Page 58

The value of sin ( \[{45}^° + \theta) - \cos ( {45}^°- \theta)\] is equal to 

  • 2 cos \[\theta\]

  • 0  

  •   2 sin \[\theta\]

  • 1

Q 30 | Page 58

The value of sin ( \[{45}^° + \theta) - \cos ( {45}^°- \theta)\] is equal to 

  • 2 cos \[\theta\]

  • 0  

  •   2 sin \[\theta\]

  • 1

Q 31 | Page 58

If  \[∆ ABC\]is right angled at C , then the value of cos ( \[A + B\]) is

Q 31 | Page 58

If  \[∆ ABC\]is right angled at C , then the value of cos ( \[A + B\]) is

Q 32 | Page 59

If cos  \[9\theta\] = sin \[\theta\] and  \[9\theta\]  < 900 , then the value of tan \[6 \theta\] is

Q 32 | Page 59

If cos  \[9\theta\] = sin \[\theta\] and  \[9\theta\]  < 900 , then the value of tan \[6 \theta\] is

Q 33 | Page 59

If  cos (\[\alpha + \beta\]= 0 , then sin \[\left( \alpha - \beta \right)\] can be reduced to  

 

Q 33 | Page 59

If  cos (\[\alpha + \beta\]= 0 , then sin \[\left( \alpha - \beta \right)\] can be reduced to  

 

Chapter 11: Trigonometric Identities

Ex. 11.10Ex. 11.20Others

RD Sharma 10 Mathematics

10 Mathematics - Shaalaa.com

RD Sharma solutions for Class 10 Mathematics chapter 11 - Trigonometric Identities

RD Sharma solutions for Class 10 Maths chapter 11 (Trigonometric Identities) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 10 Mathematics chapter 11 Trigonometric Identities are Trigonometry Ratio of Zero Degree and Negative Angles, Application of Trigonometry, Heights and Distances, Trigonometric Ratios of Complementary Angles, Trigonometric Identities, Trigonometric Ratios in Terms of Coordinates of Point, Angles in Standard Position, Trigonometric Ratios of Complementary Angles, Trigonometric Identities.

Using RD Sharma Class 10 solutions Trigonometric Identities exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 10 prefer RD Sharma Textbook Solutions to score more in exam.

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