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

Prove the following trigonometric identities:

(1 – cos² A) cosec² A = 1

Prove the following trigonometric identities:

(1 + cot² A) sin² A = 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^2 A + 1/(1 + cot^2 A) = 1`

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:

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

Prove the following trigonometric identities.

tan² θ − sin² θ = tan² θ sin² θ

Prove the following trigonometric identities:

(1 + tan² θ) (1 − sin θ) (1 + sin θ) = 1

Prove the following trigonometric identities:

sin² A cot² A + cos² A tan² A = 1

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 that: `(1 + "cosec"  θ)/("cosec"  θ) = (cos^2 θ)/(1 - sin θ)`

Prove the following trigonometric identities.

sec⁶ θ = tan⁶ θ + 3 tan² θ sec² θ + 1

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.

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

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)) + sqrt((1 + cos A)/(1 - cos A)) = 2  "cosec"  A`

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 the following trigonometric identities:

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

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.

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

Prove the following trigonometric identities:

tan² A + cot² A = sec² A cosec² A − 2

Prove that:

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

Prove the following trigonometric identities.

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

Prove that:

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

Prove the following trigonometric identities.

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

Prove the following trigonometric identities.

sec A (1 − sin A) (sec A + tan A) = 1

Prove the following trigonometric identities.

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

Prove that:

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

Prove the following trigonometric identities.

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

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

If x cos θ + y sin θ = a and x sin θ – y cos θ = b, prove that a² + b² = x² + y².

Use the identity sin² A + cos² A = 1 to prove that tan² A + 1 = sec² A. Hence, find the value of tan A when sec A = `5/3`, where A is an acute angle.

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

If x cos θ + y sin θ = a and x sin θ – y cos θ = b, prove that a² + b² = x² + y².

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

Prove the following identities:

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

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 identity (sin θ + cos θ)(tan θ + cot θ) = sec θ + cosec θ.

Prove that `(1 + sec θ - tan θ)/(1 + sec θ + tan θ) = (1 - sin θ)/(cos θ)`.

Prove the following identities:

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

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.

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

Prove the following trigonometric identities.

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

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

If sin θ + 2 cos θ = 1 prove that 2 sin θ – cos θ = 2.

If T_n = sinⁿ θ + cosⁿ θ, prove that `(T_3 - T_5)/(T_1) = (T_5 - T_7)/(T_3)`

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) = 2a^(2/3)`

If x = a cos³θ and y = b sin³θ, prove that `(x/a)^(2/3) + (y/b)^(2/3) = 1`.

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

Prove the following trigonometric identities.

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

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 β) (1 − cos γ)

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

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 - z^2/c^2 = 1`

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 4 tan θ = 3, evaluate `((4sin theta - cos theta + 1)/(4sin theta + cos theta - 1))`

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`

If `"cosec"  θ = 13/12`, find the value of `(2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ)`

If `sin θ + cos θ = sqrt(2) sin θ`, find cot θ.

If 2sin²θ – cos²θ = 2, then find the value of θ.

If `sqrt(3) tan θ - 1 = 0`, find the value of sin²θ – cos²θ.

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 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. 

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 \[\cos \theta = \frac{3}{4}\] then find the value of 9 tan² θ + 9. 

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 θ.

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 \[5\left( x^2 - \frac{1}{x^2} \right)\] 

Evaluate the following:

`(3 sin 30° - 4 sin^3 30°)/(2 sin^2 50° + 2 cos^2 50°)`

Find the value of x for which (sin A + cosec A)² + (cos A + sec A)² = x + tan² A + cot² A.

If sin A = y, then express cos A and tan A is terms of y.

If `cosec  theta = 2x and cot theta = 2/x ," find the value of"  2 ( x^2 - 1/ (x^2))`

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

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

The value of `(sin^4θ - cos^4θ)/(sin^2θ - cos^2θ)` is ______.

If `1/(1 + sin θ) + 1/(1 - sin θ) = k sec^2θ`, then the value of k is ______.

If `sqrt((1 - cos^2 θ)sec^2 θ) = k tan θ` and 0 < θ < 90°, then k = ______.

If cosec θ + cot θ = 3, then cos θ = ______.

If cosec θ – cot θ = 2, then find the value of cosec² + cot² θ is ______.

If sec θ + tan θ = m, then the value of sec⁴ θ – tan⁴ θ – 2 sec θ tan θ is ______.

The value of `(cos^4theta + cos^2theta sin^2 theta + sin^2 theta)/(cos^2 theta + cos^2 theta sin^2 theta + sin^4 theta)` is ______.

If `sin θ - cos θ = 3/5`, then sin θ cos θ = ______.

If `(sin^2θ - 3sinθ + 2)/(cos^2θ) = 1`, then θ = ______.

If cos θ + cos² θ = 1, then sin² θ + sin⁴ θ = ______.

If `(tan^3θ - 1)/(tan θ - 1) = A sec^2 θ + B tan θ`, then A + B = ______.

The value of (cosec θ – sin θ) (sec θ – cos θ) (tan θ + cot θ) is ______.

`sqrt(-4 + sqrt(8 + 16  "cosec"^4 θ + sin^4 θ)) = A  "cosec"  θ + B sin θ`, then A = ______ and B = ______.

If (tan θ + 2) (2 tan θ + 1) = A tan θ + B sec² θ, then AB = ______.

If 2 sin θ + 3 cos θ = 2, then 3 sin θ – 2 cos θ = ______.

If sin⁴ A – cos⁴ A = 1 and 0 < A ≤ 90°, then A = ______.