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# RD Sharma solutions for Class 10th Board Exam Mathematics chapter 6 - Trigonometric Identities

## Chapter 6 : Trigonometric Identities

#### Pages 43 - 47

Q 1 | Page 43

Prove the following trigonometric identities:

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

Q 1 | Page 43

Prove the following trigonometric identities:

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

Q 2 | Page 43

Prove the following trigonometric identities

(1 + cot2 A) sin2 A = 1

Q 2 | Page 43

Prove the following trigonometric identities

(1 + cot2 A) sin2 A = 1

Q 3 | Page 43

Prove the following trigonometric identities.

tan2θ cos2θ = 1 − cos2θ

Q 3 | Page 43

Prove the following trigonometric identities.

tan2θ cos2θ = 1 − cos2θ

Q 4 | Page 43

Prove the following trigonometric identities.

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

Q 4 | Page 43

Prove the following trigonometric identities.

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

Q 5 | Page 43

Prove the following trigonometric identities.

(sec2 θ − 1) (cosec2 θ − 1) = 1

Q 5 | Page 43

Prove the following trigonometric identities.

(sec2 θ − 1) (cosec2 θ − 1) = 1

Q 6 | Page 43

Prove the following trigonometric identities.

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

Q 6 | Page 43

Prove the following trigonometric identities.

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

Q 7 | Page 43

Prove the following trigonometric identities

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

Q 7 | Page 43

Prove the following trigonometric identities

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

Q 8 | Page 43

Prove the following trigonometric identities.

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

Q 8 | Page 43

Prove the following trigonometric identities.

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

Q 9 | Page 43

Prove the following trigonometric identities

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

Q 9 | Page 43

Prove the following trigonometric identities

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

Q 10 | Page 43

Prove the following trigonometric identities.

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

Q 10 | Page 43

Prove the following trigonometric identities.

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

Q 11 | Page 43

Prove the following trigonometric identities.

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

Q 11 | Page 43

Prove the following trigonometric identities.

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

Q 12 | Page 43

Prove the following trigonometric identities.

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

Q 12 | Page 43

Prove the following trigonometric identities.

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

Q 13 | Page 44

Prove the following trigonometric identities.

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

Q 13 | Page 44

Prove the following trigonometric identities.

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

Q 14 | Page 44

Prove the following trigonometric identities.

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

Q 14 | Page 44

Prove the following trigonometric identities.

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

Q 15 | Page 44

Prove the following trigonometric identities.

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

Q 15 | Page 44

Prove the following trigonometric identities.

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

Q 16 | Page 44

Prove the following trigonometric identities.

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

Q 16 | Page 44

Prove the following trigonometric identities.

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

Q 17 | Page 44

Prove the following trigonometric identities.

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

Q 17 | Page 44

Prove the following trigonometric identities.

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

Q 18 | Page 44

Prove the following trigonometric identities.

secA (1 − sinA) (secA + tanA) = 1

Q 18 | Page 44

Prove the following trigonometric identities.

secA (1 − sinA) (secA + tanA) = 1

Q 19 | Page 44

Prove the following trigonometric identities.

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

Q 19 | Page 44

Prove the following trigonometric identities.

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

Q 20 | Page 44

Prove the following trigonometric identities.

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

Q 20 | Page 44

Prove the following trigonometric identities.

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

Q 21 | Page 44

Prove the following trigonometric identities.

(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1

Q 21 | Page 44

Prove the following trigonometric identities.

(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1

Q 22 | Page 44

Prove the following trigonometric identities.

sin2 A cot2 A + cos2 A tan2 A = 1

Q 22 | Page 44

Prove the following trigonometric identities.

sin2 A cot2 A + cos2 A tan2 A = 1

Q 23.1 | Page 44

Prove the following trigonometric identities.

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

Q 23.1 | Page 44

Prove the following trigonometric identities.

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

Q 23.2 | Page 44

Prove the following trigonometric identities.

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

Q 23.2 | Page 44

Prove the following trigonometric identities.

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

Q 24 | Page 44

Prove the following trigonometric identities.

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

Q 24 | Page 44

Prove the following trigonometric identities.

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

Q 25 | Page 44

Prove the following trigonometric identities.

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

Q 25 | Page 44

Prove the following trigonometric identities.

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

Q 26 | Page 44

Prove the following trigonometric identities.

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

Q 26 | Page 44

Prove the following trigonometric identities.

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

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)

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)

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

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

Q 29 | Page 44

Prove the following trigonometric identities.

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

Q 29 | Page 44

Prove the following trigonometric identities.

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

Q 30 | Page 44

Prove the following trigonometric identities.

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

Q 30 | Page 44

Prove the following trigonometric identities.

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

Q 31 | Page 44

Prove the following trigonometric identities.

sec6θ = tan6θ + 3 tan2θ sec2θ + 1

Q 31 | Page 44

Prove the following trigonometric identities.

sec6θ = tan6θ + 3 tan2θ sec2θ + 1

Q 32 | Page 44

Prove the following trigonometric identities

cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1

Q 32 | Page 44

Prove the following trigonometric identities

cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1

Q 33 | Page 44

Prove the following trigonometric identities.

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

Q 33 | Page 44

Prove the following trigonometric identities.

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

Q 34 | Page 44

Prove the following trigonometric identities.

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

Q 34 | Page 44

Prove the following trigonometric identities.

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

Q 35 | Page 44

Prove the following trigonometric identities.

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

Q 35 | Page 44

Prove the following trigonometric identities.

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

Q 36 | Page 44

Prove the following trigonometric identities.

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

Q 36 | Page 44

Prove the following trigonometric identities.

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

Q 37 | Page 44

Prove the following trigonometric identities.

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

Q 37 | Page 44

Prove the following trigonometric identities.

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

Q 38 | Page 44

Prove the following trigonometric identities.

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

Q 38 | Page 44

Prove the following trigonometric identities.

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

Q 39 | Page 45

Prove the following trigonometric identities.

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

Q 39 | Page 45

Prove the following trigonometric identities.

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

Q 40 | Page 45

Prove the following trigonometric identities.

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

Q 40 | Page 45

Prove the following trigonometric identities.

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

Q 41 | Page 45

Prove the following trigonometric identities.

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

Q 41 | Page 45

Prove the following trigonometric identities.

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

Q 42 | Page 45

Prove the following trigonometric identities.

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

Q 42 | Page 45

Prove the following trigonometric identities.

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

Q 43 | Page 45

Prove the following trigonometric identities.

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

Q 43 | Page 45

Prove the following trigonometric identities.

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

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)

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)

Q 45 | Page 45

Prove the following trigonometric identities.

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

Q 45 | Page 45

Prove the following trigonometric identities.

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

Q 46 | Page 45

Prove the following trigonometric identities.

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

Q 46 | Page 45

Prove the following trigonometric identities.

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

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

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

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)

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)

Q 47.3 | Page 45

Prove the following trigonometric identities.

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

Q 47.3 | Page 45

Prove the following trigonometric identities.

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

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)

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)

Q 49 | Page 45

Prove the following trigonometric identities

tan2 A + cot2 A = sec2 A cosec2 A − 2

Q 49 | Page 45

Prove the following trigonometric identities

tan2 A + cot2 A = sec2 A cosec2 A − 2

Q 50 | Page 45

Prove the following trigonometric identities.

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

Q 50 | Page 45

Prove the following trigonometric identities.

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

Q 51 | Page 45

Prove the following trigonometric identities.

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

Q 51 | Page 45

Prove the following trigonometric identities.

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

Q 51 | Page 45

Prove the following trigonometric identities.

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

Q 51 | Page 45

Prove the following trigonometric identities.

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

Q 52 | Page 45

Prove the following trigonometric identities.

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

Q 52 | Page 45

Prove the following trigonometric identities.

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

Q 53 | Page 45

Prove the following trigonometric identities.

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

Q 53 | Page 45

Prove the following trigonometric identities.

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

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

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

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

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

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))

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))

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)

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)

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)

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)

Q 59 | Page 45

Prove the following trigonometric identities.

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

Q 59 | Page 45

Prove the following trigonometric identities.

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

Q 60 | Page 46

Prove the following trigonometric identities.

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

Q 60 | Page 46

Prove the following trigonometric identities.

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

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)

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)

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

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

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

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

Q 64 | Page 46

Prove the following trigonometric identities.

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

Q 64 | Page 46

Prove the following trigonometric identities.

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

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

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

Q 66 | Page 46

Prove the following trigonometric identities

sec4 A(1 − sin4 A) − 2 tan2 A = 1

Q 66 | Page 46

Prove the following trigonometric identities

sec4 A(1 − sin4 A) − 2 tan2 A = 1

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))

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))

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

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

Q 69 | Page 46

Prove the following trigonometric identities.

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

Q 69 | Page 46

Prove the following trigonometric identities.

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

Q 70 | Page 46

Prove the following trigonometric identities.

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

Q 70 | Page 46

Prove the following trigonometric identities.

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

Q 71 | Page 46

Prove the following trigonometric identities.

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

Q 71 | Page 46

Prove the following trigonometric identities.

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

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

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

Q 73 | Page 46

Prove the following trigonometric identities.

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

Q 73 | Page 46

Prove the following trigonometric identities.

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

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

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

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

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

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

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

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 = nProve that (m + n)^(2/3) + (m - n)^(2/3)

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 = nProve that (m + n)^(2/3) + (m - n)^(2/3)

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

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

Q 79 | Page 47

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

Q 79 | Page 47

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

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

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

Q 81 | Page 47

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

Q 81 | Page 47

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

Q 82 | Page 47

Prove the following trigonometric identities.

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

Q 82 | Page 47

Prove the following trigonometric identities.

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

Q 83.1 | Page 47

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

Q 83.1 | Page 47

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

Q 83.2 | Page 47

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

Q 83.2 | Page 47

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

Q 83.3 | Page 47

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

Q 83.3 | Page 47

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

Q 83.4 | Page 47

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

Q 83.4 | Page 47

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

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

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

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

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

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

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

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

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

#### Page 54

Q 1 | Page 54

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

Q 1 | Page 54

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

Q 2 | Page 54

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

Q 2 | Page 54

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

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)

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)

Q 4 | Page 54

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

Q 4 | Page 54

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

Q 5 | Page 54

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

Q 5 | Page 54

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

Q 6 | Page 54

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

Q 6 | Page 54

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

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))

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))

Q 8 | Page 54

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

Q 8 | Page 54

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

Q 9 | Page 54

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

Q 9 | Page 54

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

Q 10 | Page 54

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

Q 10 | Page 54

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

## RD Sharma solutions for Class 10th Board Exam Mathematics chapter 6 - Trigonometric Identities

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Concepts covered in Class 10th Board Exam Mathematics chapter 6 Trigonometric Identities are 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, Trigonometry Ratio of Zero Degree and Negative Angles, Trigonometric Identities, Trigonometric Ratios of Complementary Angles.

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