Nootan solutions for मैथमैटिक्स [अंग्रेजी] कक्षा १० आईसीएसई chapter 18 - Trigonometric identities [Latest edition]

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:

`1/(tan⁡ theta + cot theta) = sin theta.cos⁡ theta`

cosec⁴ θ − cosec² θ = cot⁴ θ + cot² θ

Prove the following identities:

(cos A + sin A)² + (cos A – sin A)² = 2

Prove the following:

cot² θ − cos² θ = cot² θ.cos²θ

Prove the following:

`1/(1 + sinA) - 1/(1 - sinA) = - 2secA tanA`

Prove the following:

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

Prove the following trigonometric identities.

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

Prove the following identities:

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

Prove the following:

`1/(sinθ + cosθ) + 1/(sinθ - cosθ) = (2 sinθ)/(2sin^2θ - 1)`

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

Prove that: `sqrt((1 - cos θ)/(1 + cos θ)) = "cosec" θ - cot θ`.

Prove that: `(sin θ - 2sin^3 θ)/(2 cos^3 θ - cos θ) = tan θ`.

Prove the following:

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

Prove the following:

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

`1 + (tan^2 θ)/((1 + sec θ)) = sec θ`

Prove the following:

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

Prove that: `(sec θ - tan θ)/(sec θ + tan θ ) = 1 - 2 sec θ.tan θ + 2 tan^2θ`

Prove the following trigonometric identities.

sec⁶ θ = tan⁶ θ + 3 tan² θ sec² θ + 1

Prove the following trigonometric identities.

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

`(cos^3 θ + sin^3 θ)/(cos θ + sin θ) + (cos ^3 θ - sin^3 θ)/(cos θ - sin θ) = 2`

Prove the following trigonometric identities.

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

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

Prove the following:

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

Prove the following trigonometric identities.

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

If tan A = n tan B and sin A = m sin B, prove that `cos^2A = (m^2 - 1)/(n^2 - 1)`

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

If sin θ + cos θ = m and sec θ + cosec θ = n, prove that n(m² – 1) = 2m

If tan⁴ θ + tan² θ = 1, prove that: cos⁴ + cos² θ = 1

If x = h + a cos θ, y = k + a sin θ, prove that: (x − h)² + (y − k)² = a²

Factorize: sin³θ + cos³θ

Hence, prove the following identity:

`(sin^3θ + cos^3θ)/(sin θ + cos θ) + sin θ cos θ = 1`

(1 + sin A)(1 – sin A) is equal to ______.

`(sinA.tanA)/(1 - cos A) - 1` is equal to ______.

2 sin² A + cos⁴ A – 1 is equal to ______.

If x = r cos θ, y = r sin θ then x² + y² is equal to ______.

`sqrt(sec^2 θ + cocec^2 θ)` is equal to ______.

(1 + cot² θ) is equal to ______.

If sec θ + tan θ = k, then sec θ − tan θ is equal to ______.

`(1 + tan θ)/(1 + cot θ)` is equal to ______.

(sec θ + tan θ) (1 – sin θ) is equal to ______.

Which of the following is true?

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. sin² θ + cos² θ = 1
2. cosec² θ + cot² θ = 1

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

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

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. `(1 + tan θ)/(1 + cot θ) = cot θ`
2. sec² θ − tan² θ = 1

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

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

Prove the following identities:

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

Prove the following trigonometric identities.

tan² θ − sin² θ = tan² θ sin² θ

Prove that:
`sqrt(( secθ - 1)/(secθ + 1)) + sqrt((secθ + 1)/(secθ - 1)) = 2 "cosec"θ`

If a cos θ – b sin θ = c, then prove that (a sin θ + b cos θ) = `±  sqrt(a^2 + b^2 - c^2)`

Prove the following trigonometric identities.

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

If sinθ + sin² θ = 1, prove that cos² θ + cos⁴ θ = 1

Prove that:

tan² θ + cot² θ = sec² θ cosec² θ − 2

If sec θ + tan θ = p, prove that: sin θ = `(p^2 - 1)/(p^2 + 1)`

Prove the following identities.

cot θ + tan θ = sec θ cosec θ

Prove that: `sqrt((1 - cos θ)/(1 + cos θ)) = "cosec" θ - cot θ`.