NCERT solutions for मैथमैटिक्स पार्ट १ एण्ड २ [अंग्रेजी] कक्षा १२ chapter 2 - Inverse Trigonometric Functions [Latest edition]

Find the principal value of the following:

`sin^(-1) (-1/2)`

Find the principal value of the following:

`cos^(-1) (sqrt3/2)`

Find the principal value of the following:

cosec⁻¹ (2)

Find the principal value of the following:

`tan^(-1) (-sqrt3)`

Find the principal value of the following:

`cos^(-1) (-1/2)`

Find the principal value of the following:

tan⁻¹ (−1)

Find the principal value of the following:

`sec^(-1) (2/sqrt(3))`

Find the principal value of the following:

`cot^(-1) (sqrt3)`

Find the principal value of the following:

`cos^(-1) (-1/sqrt2)`

Find the principal value of the following:

`"cosec"^(-1)(-sqrt2)`

Find the value of the following:

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

Find the value of the following:

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

If sin⁻¹ x = y, then ______.

`tan^(-1) sqrt3 - sec^(-1)(-2)` is equal to ______.

Prove the following:

3 sin⁻¹ x = sin⁻¹ (3x − 4x³), `x ∈ [-1/2, 1/2]`

Prove the following: 

3cos⁻¹x = cos⁻¹(4x³ − 3x), `x ∈ [1/2, 1]`

Prove `tan^(-1)  2/11 + tan^(-1)  7/24 = tan^(-1)  1/2`

Prove `2 tan^(-1)  1/2 + tan^(-1)  1/7 = tan^(-1)  31/17`

Write the following function in the simplest form:

`tan^(-1)  (sqrt(1+x^2) -1)/x`, x ≠ 0

Write the function in the simplest form: `tan^(-1)  1/(sqrt(x^2 - 1)), |x| > 1`

Write the following function in the simplest form:

`tan^(-1) (sqrt((1-cos x)/(1 + cos x)))`, 0 < x < π

Write the function in the simplest form:  `tan^(-1)  ((cos x - sin x)/(cos x + sin x)) `,` 0 < x < pi`

Write the following function in the simplest form:

`tan^(-1)  x/(sqrt(a^2 - x^2))`, |x| < a

Write the following function in the simplest form:

`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)), a > 0; (-a)/sqrt3 < x < a/sqrt3`

Find the value of the following:

`tan^-1 [2 cos (2  sin^-1  1/2)]`

Find the value of `cot(tan^(-1) a + cot^(-1) a)`

Find the value of the following:

`tan  1/2 [sin^(-1)  (2x)/(1+ x^2) + cos^(-1)  (1-y^2)/(1+y^2)]`, |x| < 1, y > 0 and xy < 1

if `sin(sin^(-1)  1/5 + cos^(-1) x)  = 1` then find the value of x

if `tan^(-1)  (x-1)/(x - 2) + tan^(-1)  (x + 1)/(x + 2) = pi/4` then find the value of x.

Find the value of the given expression.

`sin^(-1) (sin  (2pi)/3)`

Find the value of the given expression.

`tan^(-1) (tan  (3pi)/4)`

Find the value of the given expression.

`tan(sin^(-1)  3/5 + cot^(-1)  3/2)`

`cos^(-1) (cos  (7pi)/6)` is equal to ______.

`sin[pi/3 - sin^(-1) (-1/2)]` is equal to ______.

Find the value of the following:

`cos^(-1) (cos  (13pi)/6)`

Find the value of the following:

`tan^(-1) (tan  (7pi)/6)`

Show that `2sin^-1(3/5) = tan^-1(24/7)`

Prove that:

`sin^(-1)  8/17 + sin^(-1)  3/5 = tan^(-1)  77/36`

Prove that:

`cos^(-1)  4/5 + cos^(-1)  12/13 = cos^(-1)  33/65`

Prove that:

`cos^(-1)  12/13 + sin^(-1)  3/5 = sin^(-1)  56/65`

Prove that:

`tan^(-1)  63/16 = sin^(-1)  5/13 + cos^(-1)  3/5`

Prove `tan^(-1)   1/5 + tan^(-1)  (1/7) + tan^(-1)  1/3 + tan^(-1)  1/8 = pi/4`

Prove that:

`tan^(-1) sqrtx = 1/2 cos^(-1)  (1-x)/(1+x)`, x ∈ [0, 1]

Prove that:

`cot^(-1)  ((sqrt(1+sin x) + sqrt(1-sinx))/(sqrt(1+sin x) - sqrt(1- sinx))) = x/2, x in (0, pi/4)`

Prove that:

`tan^-1 ((sqrt(1 + x) - sqrt(1 - x))/(sqrt(1 + x) + sqrt(1 - x))) = pi/4 - 1/2 cos^-1 x`, for `- 1/sqrt2 ≤ x ≤ 1`

[Hint: Put x =  cos 2θ]

Prove `(9pi)/8 - 9/4  sin^(-1)  1/3 = 9/4 sin^(-1)  (2sqrt2)/3`

Solve the following equation:

2 tan⁻¹ (cos x) = tan⁻¹ (2 cosec x)

Solve the following equation for x:

`tan^-1((1-x)/(1+x))-1/2 tan^-1x` = 0, where x > 0

sin (tan^–1 x), |x| < 1 is equal to ______.

sin^–1 (1 – x) – 2 sin^–1 x = `pi/2`, then x is equal to ______.

Solve  `tan^(-1) -  tan^(-1)  (x - y)/(x+y)` is equal to

(A) `pi/2`

(B). `pi/3` 

(C) `pi/4` 

(D) `(-3pi)/4`