RD Sharma solutions for Mathematics [English] Class 12 chapter 4 - Inverse Trigonometric Functions [Latest edition]

Find the principal value of the following:

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

Find the principal value of the following:

`sin^-1(cos  (2pi)/3)`

Find the principal value of the following:

`sin^-1((sqrt3-1)/(2sqrt2))`

Find the principal value of the following:

`sin^-1((sqrt3+1)/(2sqrt2))`

Find the principal value of the following:

`sin^-1(cos  (3pi)/4)`

Find the principal value of the following:

`sin^-1(tan  (5pi)/4)`

`sin^-1  1/2-2sin^-1  1/sqrt2`

`sin^-1{cos(sin^-1  sqrt3/2)}`

Find the domain of the following function:

`f(x)=sin^-1x^2`

Find the domain of the following function:

`f(x) = sin^-1x + sinx`

Find the domain of the following function:

`f(x)sin^-1sqrt(x^2-1)`

Find the domain of the following function:

`f(x)=sin^-1x+sin^-1 2x`

If `sin^-1 x + sin^-1 y+sin^-1 z+sin^-1 t=2pi` , then find the value of x² + y² + z² + t² 

If `(sin^-1x)^2 + (sin^-1y)^2+(sin^-1z)^2=3/4pi^2,`  find the value of x² + y² + z² 

Find the domain of definition of `f(x)=cos^-1(x^2-4)`

Find the domain of  `f(x) =2cos^-1 2x+sin^-1x.`

Find the domain of `f(x)=cos^-1x+cosx.`

​Find the principal values of the following:
`cos^-1(-sqrt3/2)`

​Find the principal values of the following:

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

​Find the principal values of the following:

`cos^-1(sin   (4pi)/3)`

​Find the principal values of the following:

`cos^-1(tan  (3pi)/4)`

For the principal value, evaluate of the following:

`cos^-1  1/2+2sin^-1  (1/2)`

For the principal value, evaluate of the following:

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

For the principal value, evaluate of the following:

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

Find the principal value of the following:

`tan^-1(1/sqrt3)`

Find the principal value of the following:

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

Find the principal value of the following:

`tan^-1(cos  pi/2)`

Find the principal value of the following:

`tan^-1(2cos  (2pi)/3)`

For the principal value, evaluate of the following:

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

For the principal value, evaluate of the following:

`tan^-1{2sin(4cos^-1  sqrt3/2)}`

Evaluate the following:

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

Evaluate the following:

`tan^-1(-1/sqrt3)+tan^-1(-sqrt3)+tan^-1(sin(-pi/2))`

Evaluate the following:

`tan^-1(tan  (5pi)/6)+cos^-1{cos((13pi)/6)}`

Find the principal value of the following:

`sec^-1(-sqrt2)`

Find the principal value of the following:

`sec^-1(2)`

Find the principal value of the following:

`sec^-1(2sin  (3pi)/4)`

Find the principal value of the following:

`sec^-1(2tan  (3pi)/4)`

For the principal value, evaluate the following:

`tan^-1sqrt3-sec^-1(-2)`

For the principal value, evaluate the following:

`sin^-1(-sqrt3/2)-2sec^-1(2tan  pi/6)`

Find the domain of `sec^(-1)(3x-1)`.

Find the domain of `sec^(-1) x-tan^(-1)x`

​Find the principal value of the following:

`cosec^-1(-sqrt2)`

​Find the principal value of the following:

cosec^-1(-2)

​Find the principal value of the following:

`\text(cosec)^-1(2/sqrt3)`

​Find the principal value of the following:

`cosec^-1(2cos  (2pi)/3)`

Find the set of values of `cosec^-1(sqrt3/2)`

For the principal value, evaluate the following:

`sin^-1(-sqrt3/2)+\text{cosec}^-1(-2/sqrt3)`

For the principal value, evaluate the following:

`sec^-1(sqrt2)+2\text{cosec}^-1(-sqrt2)`

For the principal value, evaluate the following:

`sin^-1[cos{2\text(cosec)^-1(-2)}]`

For the principal value, evaluate the following:

`cosec^-1(2tan  (11pi)/6)`

Find the principal value of the following:

`cot^-1(-sqrt3)`

Find the principal value of the following:

`cot^-1(sqrt3)`

Find the principal value of the following:

`cot^-1(-1/sqrt3)`

Find the principal value of the following:

`cot^-1(tan  (3pi)/4)`

Find the domain of `f(x)=cotx+cot^-1x`

Evaluate the following:

`cot^-1  1/sqrt3-\text(cosec)^-1(-2)+sec^-1(2/sqrt3)`

Evaluate the following:

`cot^-1{2cos(sin^-1  sqrt3/2)}`

Evaluate the following:

`\text(cosec)^-1(-2/sqrt3)+2cot^-1(-1)`

Evaluate the following:

`tan^-1(-1/sqrt3)+cot^-1(1/sqrt3)+tan^-1(sin(-pi/2))`

`sin^-1(sin  pi/6)`

`sin^-1(sin  (7pi)/6)`

`sin^-1(sin  (5pi)/6)`

`sin^-1(sin  (13pi)/7)`

`sin^-1(sin  (17pi)/8)`

`sin^-1{(sin - (17pi)/8)}`

`sin^-1(sin3)`

`sin^-1(sin4)`

`sin^-1(sin12)`

`sin^-1(sin2)`

Evaluate the following:

`cos^-1{cos(-pi/4)}`

Evaluate the following:

`cos^-1{cos  (5pi)/4}`

Evaluate the following:

`cos^-1{cos  ((4pi)/3)}`

Evaluate the following:

`cos^-1{cos  (13pi)/6}`

Evaluate the following:

`cos^-1(cos3)`

Evaluate the following:

`cos^-1(cos4)`

Evaluate the following:

`cos^-1(cos5)`

Evaluate the following:

`cos^-1(cos12)`

Evaluate the following:

`tan^-1(tan  pi/3)`

Evaluate the following:

`tan^-1(tan  (6pi)/7)`

Evaluate the following:

`tan^-1(tan  (7pi)/6)`

Evaluate the following:

`tan^-1(tan  (9pi)/4)`

Evaluate the following:

`tan^-1(tan1)`

Evaluate the following:

`tan^-1(tan2)`

Evaluate the following:

`tan^-1(tan4)`

Evaluate the following:

`tan^-1(tan12)`

Evaluate the following:

`sec^-1(sec  pi/3)`

Evaluate the following:

`sec^-1(sec  (2pi)/3)`

Evaluate the following:

`sec^-1(sec  (5pi)/4)`

Evaluate the following:

`sec^-1(sec  (7pi)/3)`

Evaluate the following:

`sec^-1(sec  (9pi)/5)`

Evaluate the following:

`sec^-1{sec  (-(7pi)/3)}`

Evaluate the following:

`sec^-1(sec  (13pi)/4)`

Evaluate the following:

`sec^-1(sec  (25pi)/6)`

Evaluate the following:

`\text(cosec)^-1(\text{cosec}  pi/4)`

Evaluate the following:

`cosec^-1(cosec  (3pi)/4)`

Evaluate the following:

`cosec^-1(cosec  (6pi)/5)`

Evaluate the following:

`cosec^-1(cosec  (11pi)/6)`

Evaluate the following:

`cosec^-1(cosec  (13pi)/6)`

Evaluate the following:

`cosec^-1{cosec  (-(9pi)/4)}`

Evaluate the following:

`cot^-1(cot  pi/3)`

Evaluate the following:

`cot^-1(cot  (4pi)/3)`

Evaluate the following:

`cot^-1(cot  (9pi)/4)`

Evaluate the following:

`cot^-1(cot  (19pi)/6)`

Evaluate the following:

`cot^-1{cot (-(8pi)/3)}`

Evaluate the following:

`cot^-1{cot  ((21pi)/4)}`

Write the following in the simplest form:

`cot^-1  a/sqrt(x^2-a^2),|  x  | > a`

Write the following in the simplest form:

`tan^-1{x+sqrt(1+x^2)},x in R `

Write the following in the simplest form:

`tan^-1{sqrt(1+x^2)-x},x in R `

Write the following in the simplest form:

`tan^-1{(sqrt(1+x^2)-1)/x},x !=0`

Write the following in the simplest form:

`tan^-1{(sqrt(1+x^2)+1)/x},x !=0`

Write the following in the simplest form:

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

Write the following in the simplest form:

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

Write the following in the simplest form:

`sin^-1{(x+sqrt(1-x^2))/sqrt2},-1<x<1`

Write the following in the simplest form:

`sin^-1{(sqrt(1+x)+sqrt(1-x))/2},0<x<1`

Write the following in the simplest form:

`sin{2tan^-1sqrt((1-x)/(1+x))}`

Evaluate the following:

`sin(sin^-1  7/25)`

Evaluate the following:

`sin(cos^-1  5/13)`

Evaluate the following:

`sin(tan^-1  24/7)`

Evaluate the following:

`sin(sec^-1  17/8)`

Evaluate the following:

`cosec(cos^-1  3/5)`

Evaluate the following:

`sec(sin^-1  12/13)`

Evaluate the following:

`tan(cos^-1  8/17)`

Evaluate the following:

`cot(cos^-1  3/5)`

Evaluate the following:

`cos(tan^-1  24/7)`

Prove the following result

`tan(cos^-1  4/5+tan^-1  2/3)=17/6`

Prove the following result

`cos(sin^-1  3/5+cot^-1  3/2)=6/(5sqrt13)`

Prove the following result-

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

Prove the following result

`sin(cos^-1  3/5+sin^-1  5/13)=63/65`

Solve: `cos(sin^-1x)=1/6`

Evaluate:

`cos{sin^-1(-7/25)}`

Evaluate:

`sec{cot^-1(-5/12)}`

Evaluate:

`cot{sec^-1(-13/5)}`

Evaluate:

`tan{cos^-1(-7/25)}`

Evaluate:

`cosec{cot^-1(-12/5)}`

Evaluate:

`cos(tan^-1  3/4)`

Evaluate: `sin{cos^-1(-3/5)+cot^-1(-5/12)}`

Evaluate: 

`cot(sin^-1  3/4+sec^-1  4/3)`

Evaluate:

`sin(tan^-1x+tan^-1  1/x)` for x < 0

Evaluate:

`sin(tan^-1x+tan^-1  1/x)` for x > 0

Evaluate:

`cot(tan^-1a+cot^-1a)`

Evaluate:

`cos(sec^-1x+\text(cosec)^-1x)`,|x|≥1

If `cos^-1x + cos^-1y =pi/4,`  find the value of `sin^-1x+sin^-1y`

If `sin^-1x+sin^-1y=pi/3`  and  `cos^-1x-cos^-1y=pi/6`,  find the values of x and y.

If `cot(cos^-1  3/5+sin^-1x)=0`, find the values of x.

If `(sin^-1x)^2+(cos^-1x)^2=(17pi^2)/36,`  Find x

`sin(sin^-1  1/5+cos^-1x)=1`

`sin^-1x=pi/6+cos^-1x`

`4sin^-1x=pi-cos^-1x`

`tan^-1x+2cot^-1x=(2x)/3`

`5tan^-1x+3cot^-1x=2x`

Prove the following result:

`tan^-1  1/7+tan^-1  1/13=tan^-1  2/9`

Prove the following result:

`sin^-1  12/13+cos^-1  4/5+tan^-1  63/16=pi`

Prove the following result:

`tan^-1  1/4+tan^-1  2/9=sin^-1  1/sqrt5`

Find the value of `tan^-1  (x/y)-tan^-1((x-y)/(x+y))`

Solve the following equation for x:

`tan^-1  2x+tan^-1  3x = npi+(3pi)/4`

Solve the following equation for x:

tan⁻¹(x + 1) + tan⁻¹(x − 1) = tan⁻¹`8/31`

Solve the following equation for x:

tan⁻¹(x −1) + tan⁻¹x tan⁻¹(x + 1) = tan⁻¹3x

Solve the following equation for x:

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

Solve the following equation for x:

 cot⁻¹x − cot⁻¹(x + 2) =`pi/12`, x > 0

Solve the following equation for x:

tan⁻¹(x + 2) + tan⁻¹(x − 2) = tan⁻¹ `(8/79)`, x > 0

Solve the following equation for x:

`tan^-1  x/2+tan^-1  x/3=pi/4, 0<x<sqrt6`

Solve the following equation for x:

`tan^-1((x-2)/(x-4))+tan^-1((x+2)/(x+4))=pi/4`

Solve the following equation for x:

`tan^-1(2+x)+tan^-1(2-x)=tan^-1  2/3, where  x< -sqrt3 or, x>sqrt3`

Solve the following equation for x:

`tan^-1  (x-2)/(x-1)+tan^-1  (x+2)/(x+1)=pi/4`

Sum the following series:

`tan^-1  1/3+tan^-1  2/9+tan^-1  4/33+...+tan^-1  (2^(n-1))/(1+2^(2n-1))`

Evaluate: `cos(sin^-1  3/5+sin^-1  5/13)`

`sin^-1  63/65=sin^-1  5/13+cos^-1  3/5`

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

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

Solve the following:

`sin^-1x+sin^-1  2x=pi/3`

Solve the following:

`cos^-1x+sin^-1  x/2=π/6`

If `cos^-1  x/2+cos^-1  y/3=alpha,` then prove that  `9x^2-12xy cosa+4y^2=36sin^2a.`

Solve the equation `cos^-1  a/x-cos^-1  b/x=cos^-1  1/b-cos^-1  1/a`

Solve `cos^-1sqrt3x+cos^-1x=pi/2`

Prove that: `cos^-1  4/5+cos^-1  12/13=cos^-1  33/65`

Evaluate the following:

`tan{2tan^-1  1/5-pi/4}`

Evaluate the following:

`tan  1/2(cos^-1  sqrt5/3)`

Evaluate the following:

`sin(1/2cos^-1  4/5)`

Evaluate the following:

`sin(2tan^-1  2/3)+cos(tan^-1sqrt3)`

Prove that:

`2sin^-1  3/5=tan^-1  24/7`

`tan^-1  1/4+tan^-1  2/9=1/2cos^-1  3/2=1/2sin^-1(4/5)`

`tan^-1  2/3=1/2tan^-1  12/5`

`tan^-1  1/7+2tan^-1  1/3=pi/4`

`sin^-1  4/5+2tan^-1  1/3=pi/2`

`2sin^-1  3/5-tan^-1  17/31=pi/4`

`2tan^-1  1/5+tan^-1  1/8=tan^-1  4/7`

`2tan^-1  3/4-tan^-1  17/31=pi/4`

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

`4tan^-1  1/5-tan^-1  1/239=pi/4`

If `sin^-1  (2a)/(1+a^2)-cos^-1  (1-b^2)/(1+b^2)=tan^-1  (2x)/(1-x^2)`, then prove that `x=(a-b)/(1+ab)`

Prove that

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

Prove that

`sin{tan^-1  (1-x^2)/(2x)+cos^-1  (1-x^2)/(2x)}=1`

If `sin^-1  (2a)/(1+a^2)+sin^-1  (2b)/(1+b^2)=2tan^-1x,` Prove that  `x=(a+b)/(1-ab).`

Show that `2tan^-1x+sin^-1  (2x)/(1+x^2)` is constant for x ≥ 1, find that constant.

Find the value of the following:

`tan^-1{2cos(2sin^-1  1/2)}`

Find the value of the following:

`cos(sec^-1x+\text(cosec)^-1x),` | x | ≥ 1

Solve the following equation for x:

`tan^-1  1/4+2tan^-1  1/5+tan^-1  1/6+tan^-1  1/x=pi/4`

Solve the following equation for x:

`3sin^-1  (2x)/(1+x^2)-4cos^-1  (1-x^2)/(1+x^2)+2tan^-1  (2x)/(1-x^2)=pi/3`

Solve the following equation for x:

`tan^-1((2x)/(1-x^2))+cot^-1((1-x^2)/(2x))=(2pi)/3,x>0`

Solve the following equation for x:

`2tan^-1(sinx)=tan^-1(2sinx),x!=pi/2`

Solve the following equation for x:

`cos^-1((x^2-1)/(x^2+1))+1/2tan^-1((2x)/(1-x^2))=(2x)/3`

Solve the following equation for x:

`tan^-1((x-2)/(x-1))+tan^-1((x+2)/(x+1))=pi/4`

Prove that `2tan^-1(sqrt((a-b)/(a+b))tan  theta/2)=cos^-1((a costheta+b)/(a+b costheta))`

Prove that:

`tan^-1  (2ab)/(a^2-b^2)+tan^-1  (2xy)/(x^2-y^2)=tan^-1  (2alphabeta)/(alpha^2-beta^2),`   where `alpha=ax-by  and  beta=ay+bx.`

For any a, b, x, y > 0, prove that:

`2/3tan^-1((3ab^2-a^3)/(b^3-3a^2b))+2/3tan^-1((3xy^2-x^3)/(y^3-3x^2y))=tan^-1  (2alphabeta)/(alpha^2-beta^2)`

`where  alpha =-ax+by, beta=bx+ay`

Write the value of `sin^-1((-sqrt3)/2)+cos^-1((-1)/2)`

Write the difference between maximum and minimum values of  sin⁻¹ x for x ∈ [− 1, 1].

If `sin^-1x+sin^-1y+sin^-1z=(3pi)/2,`  then write the value of x + y + z.

If x > 1, then write the value of sin−1 `((2x)/(1+x^2))` in terms of tan⁻¹ x.

If x < 0, then write the value of cos⁻¹ `((1-x^2)/(1+x^2))` in terms of tan⁻¹ x.

Write the value of tan⁻¹x + tan⁻¹ `(1/x)`for x > 0.

Write the value of tan⁻¹ x + tan⁻¹ `(1/x)`  for x < 0.

What is the value of cos⁻¹ `(cos  (2x)/3)+sin^-1(sin  (2x)/3)?`

If −1 < x < 0, then write the value of `sin^-1((2x)/(1+x^2))+cos^-1((1-x^2)/(1+x^2))`

Write the value of sin (cot⁻¹ x).

Write the value of

\[\cos^{- 1} \left( \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\].

Write the range of tan⁻¹ x.

Write the value of cos⁻¹ (cos 1540°).

Write the value of sin⁻¹

\[\left( \sin( -{600}°) \right)\].

Write the value of cos\[\left( 2 \sin^{- 1} \frac{1}{3} \right)\]

Write the value of sin⁻¹ (sin 1550°).

Evaluate sin

\[\left( \frac{1}{2} \cos^{- 1} \frac{4}{5} \right)\]

Evaluate sin \[\left( \tan^{- 1} \frac{3}{4} \right)\]

Write the value of cos⁻¹ \[\left( \tan\frac{3\pi}{4} \right)\]

Write the value of cos \[\left( 2 \sin^{- 1} \frac{1}{2} \right)\]

Write the value of cos⁻¹ (cos 350°) − sin⁻¹ (sin 350°)

Write the value of cos² \[\left( \frac{1}{2} \cos^{- 1} \frac{3}{5} \right)\]

If tan⁻¹ x + tan⁻¹ y = `pi/4`,  then write the value of x + y + xy.

Write the value of cos⁻¹ (cos 6).

Write the value of sin⁻¹ \[\left( \cos\frac{\pi}{9} \right)\]

Write the value of sin \[\left\{ \frac{\pi}{3} - \sin^{- 1} \left( - \frac{1}{2} \right) \right\}\]

Write the value of tan⁻¹\[\left\{ \tan\left( \frac{15\pi}{4} \right) \right\}\]

Write the value ofWrite the value of \[2 \sin^{- 1} \frac{1}{2} + \cos^{- 1} \left( - \frac{1}{2} \right)\]

Write the value of \[\tan^{- 1} \frac{a}{b} - \tan^{- 1} \left( \frac{a - b}{a + b} \right)\]

Write the value of cos⁻¹ \[\left( \cos\frac{5\pi}{4} \right)\]

Show that \[\sin^{- 1} (2x\sqrt{1 - x^2}) = 2 \sin^{- 1} x\]

Evaluate: \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]

If \[\tan^{- 1} (\sqrt{3}) + \cot^{- 1} x = \frac{\pi}{2},\] find x.

If \[\sin^{- 1} \left( \frac{1}{3} \right) + \cos^{- 1} x = \frac{\pi}{2},\] then find x.

Write the value of \[\sin^{- 1} \left( \frac{1}{3} \right) - \cos^{- 1} \left( - \frac{1}{3} \right)\]

If 4 sin⁻¹ x + cos⁻¹ x = π, then what is the value of x?

If x < 0, y < 0 such that xy = 1, then write the value of tan⁻¹ x + tan⁻¹ y.

What is the principal value of `sin^-1(-sqrt3/2)?`

Write the principal value of `sin^-1(-1/2)`

Write the principal value of \[\cos^{- 1} \left( \cos\frac{2\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{2\pi}{3} \right)\]

Write the value of \[\tan\left( 2 \tan^{- 1} \frac{1}{5} \right)\]

Write the principal value of \[\tan^{- 1} 1 + \cos^{- 1} \left( - \frac{1}{2} \right)\]

Write the value of \[\tan^{- 1} \left\{ 2\sin\left( 2 \cos^{- 1} \frac{\sqrt{3}}{2} \right) \right\}\]

Write the principal value of `tan^-1sqrt3+cot^-1sqrt3`

Write the principal value of \[\cos^{- 1} \left( \cos680^\circ  \right)\]

Write the value of \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]

Write the value of \[\sec^{- 1} \left( \frac{1}{2} \right)\]

Write the value of \[\cos^{- 1} \left( \cos\frac{14\pi}{3} \right)\]

Write the value of \[\cos\left( \sin^{- 1} x + \cos^{- 1} x \right), \left| x \right| \leq 1\]

Wnte the value of the expression \[\tan\left( \frac{\sin^{- 1} x + \cos^{- 1} x}{2} \right), \text { when } x = \frac{\sqrt{3}}{2}\]

Write the principal value of \[\sin^{- 1} \left\{ \cos\left( \sin^{- 1} \frac{1}{2} \right) \right\}\]

The set of values of `\text(cosec)^-1(sqrt3/2)`

Write the value of  \[\tan^{- 1} \left( \frac{1}{x} \right)\]  for x < 0 in terms of `cot^-1x`

Write the value of  `cot^-1(-x)`  for all `x in R` in terms of `cot^-1(x)`

Wnte the value of\[\cos\left( \frac{\tan^{- 1} x + \cot^{- 1} x}{3} \right), \text{ when } x = - \frac{1}{\sqrt{3}}\]

If \[\cos\left( \tan^{- 1} x + \cot^{- 1} \sqrt{3} \right) = 0\] , find the value of x.

Find the value of \[2 \sec^{- 1} 2 + \sin^{- 1} \left( \frac{1}{2} \right)\]

If \[\cos\left( \sin^{- 1} \frac{2}{5} + \cos^{- 1} x \right) = 0\], find the value of x.

Find the value of \[\cos^{- 1} \left( \cos\frac{13\pi}{6} \right)\]

Find the value of \[\tan^{- 1} \left( \tan\frac{9\pi}{8} \right)\]

If \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{\sqrt{1 + x^2} + \sqrt{1 - x^2}} \right)\]  = α, then x² =

The value of tan \[\left\{ \cos^{- 1} \frac{1}{5\sqrt{2}} - \sin^{- 1} \frac{4}{\sqrt{17}} \right\}\] is

2 tan⁻¹ {cosec (tan⁻¹ x) − tan (cot⁻¹ x)} is equal to

If  \[\cos^{- 1} \frac{x}{a} + \cos^{- 1} \frac{y}{b} = \alpha, then\frac{x^2}{a^2} - \frac{2xy}{ab}\cos \alpha + \frac{y^2}{b^2} = \]

The positive integral solution of the equation
\[\tan^{- 1} x + \cos^{- 1} \frac{y}{\sqrt{1 + y^2}} = \sin^{- 1} \frac{3}{\sqrt{10}}\text{ is }\]

If sin⁻¹ x − cos⁻¹ x = `pi/6` , then x = 

sin\[\left[ \cot^{- 1} \left\{ \tan\left( \cos^{- 1} x \right) \right\} \right]\]  is equal to

The number of solutions of the equation \[\tan^{- 1} 2x + \tan^{- 1} 3x = \frac{\pi}{4}\] is

If α = \[\tan^{- 1} \left( \tan\frac{5\pi}{4} \right) \text{ and }\beta = \tan^{- 1} \left( - \tan\frac{2\pi}{3} \right)\] , then

The number of real solutions of the equation \[\sqrt{1 + \cos 2x} = \sqrt{2} \sin^{- 1} (\sin x), - \pi \leq x \leq \pi\]

If x < 0, y < 0 such that xy = 1, then tan⁻¹ x + tan⁻¹ y equals

\[\text{ If } u = \cot^{- 1} \sqrt{\tan \theta} - \tan^{- 1} \sqrt{\tan \theta}\text{ then }, \tan\left( \frac{\pi}{4} - \frac{u}{2} \right) =\]

\[\text{ If }\cos^{- 1} \frac{x}{3} + \cos^{- 1} \frac{y}{2} = \frac{\theta}{2}, \text{ then }4 x^2 - 12xy \cos\frac{\theta}{2} + 9 y^2 =\]

If α = \[\tan^{- 1} \left( \frac{\sqrt{3}x}{2y - x} \right), \beta = \tan^{- 1} \left( \frac{2x - y}{\sqrt{3}y} \right),\] 
 then α − β =

Let f (x) = `e^(cos^-1){sin(x+pi/3}.`
Then, f (8π/9) = 

\[\tan^{- 1} \frac{1}{11} + \tan^{- 1} \frac{2}{11}\]  is equal to

If \[\cos^{- 1} \frac{x}{2} + \cos^{- 1} \frac{y}{3} = \theta,\]  then 9x² − 12xy cos θ + 4y² is equal to

If tan⁻¹ 3 + tan⁻¹ x = tan⁻¹ 8, then x =

The value of \[\sin^{- 1} \left( \cos\frac{33\pi}{5} \right)\] is 

The value of \[\cos^{- 1} \left( \cos\frac{5\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{5\pi}{3} \right)\] is

sin \[\left\{ 2 \cos^{- 1} \left( \frac{- 3}{5} \right) \right\}\]  is equal to

If θ = sin⁻¹ {sin (−600°)}, then one of the possible values of θ is

If \[3\sin^{- 1} \left( \frac{2x}{1 + x^2} \right) - 4 \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + 2 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) = \frac{\pi}{3}\] is equal to

If 4 cos⁻¹ x + sin⁻¹ x = π, then the value of x is

It \[\tan^{- 1} \frac{x + 1}{x - 1} + \tan^{- 1} \frac{x - 1}{x} = \tan^{- 1}\]   (−7), then the value of x is

If \[\cos^{- 1} x > \sin^{- 1} x\], then

In a ∆ ABC, if C is a right angle, then
\[\tan^{- 1} \left( \frac{a}{b + c} \right) + \tan^{- 1} \left( \frac{b}{c + a} \right) =\]

The value of sin \[\left( \frac{1}{4} \sin^{- 1} \frac{\sqrt{63}}{8} \right)\] is

\[\cot\left( \frac{\pi}{4} - 2 \cot^{- 1} 3 \right) =\] 

If tan⁻¹ (cot θ) = 2 θ, then θ =

If \[\sin^{- 1} \left( \frac{2a}{1 - a^2} \right) + \cos^{- 1} \left( \frac{1 - a^2}{1 + a^2} \right) = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right),\text{ where }a, x \in \left( 0, 1 \right)\] , then, the value of x is

The value of  \[\sin\left( 2\left( \tan^{- 1} 0 . 75 \right) \right)\] is equal to

If x > 1, then \[2 \tan^{- 1} x + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] is equal to

The domain of  \[\cos^{- 1} \left( x^2 - 4 \right)\] is

The value of \[\tan\left( \cos^{- 1} \frac{3}{5} + \tan^{- 1} \frac{1}{4} \right)\]