RD Sharma solutions for Mathematics [English] Class 10 chapter 11 - Trigonometric Identities [Latest edition]

Prove the following trigonometric identities:

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

Prove the following trigonometric identities

(1 + cot² A) sin² A = 1

Prove the following trigonometric identities.

tan²θ cos²θ = 1 − cos²θ

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

(sec² θ − 1) (cosec² θ − 1) = 1

Prove the following trigonometric identities.

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

Prove the following trigonometric identities

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identity.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

sec A (1 − sin A) (sec A + tan A) = 1

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

tan² θ − sin² θ = tan² θ sin² θ

Prove the following trigonometric identities.

(1 + tan²θ) (1 − sinθ) (1 + sinθ) = 1

Prove the following trigonometric identities.

sin² A cot² A + cos² A tan² A = 1

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities

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

Prove the following trigonometric identities

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

sec⁶ θ = tan⁶ θ + 3 tan² θ sec² θ + 1

Prove the following trigonometric identities

cosec⁶θ = cot⁶θ + 3 cot²θ cosec²θ + 1

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identity:

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

Prove the following trigonometric identities.

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

`Prove the following trigonometric identities.

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

Prove the following trigonometric identities. `(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identity.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities

tan² A + cot² A = sec² A cosec² A − 2

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

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`

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`

Prove the following trigonometric identities.

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

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)`

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities

sec⁴ A(1 − sin⁴ A) − 2 tan² A = 1

Prove the following trigonometric identities.

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

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`

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

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

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

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`

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

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)`

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`

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

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

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

Prove the following trigonometric identities.

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

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

Prove that

`sqrt((1 + sin θ)/(1 - sin θ)) + sqrt((1 - sin θ)/(1 + sin θ)) = 2 sec θ`

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

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

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

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

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

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`

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

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

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

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

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

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

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

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

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

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

Define an identity.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 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 . 

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

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

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

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

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

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

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

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

If sec θ + tan θ = x, then sec θ =

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

\[\frac{x^2 - 1}{2x}\] is equal to 

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

sec⁴ A − sec² A is equal to

cos⁴ A − sin⁴ A is equal to ______.

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

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

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

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

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

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

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

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

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

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

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

The value of sin² 29° + sin² 61° is

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

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

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

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

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

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

9 sec² A − 9 tan² A is equal to

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

(sec A + tan A) (1 − sin A) = ______.

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

If sin θ − cos θ = 0 then the value of sin⁴θ + cos⁴θ

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

If ∆ABC is right angled at C, then the value of cos (A + B) is ______.

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

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