If \(\cot ^{-1} x=\frac{2 \pi}{5}\) for some x \(\in\) R, the value of tan^-1 x is

The domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1}\) is

If |x| \(\le\) 1, then 2 tan^-1 x-sin^-1\(\frac{2x}{1+x^2}\) is equal to

The equation \(\tan ^{-1} x-\cot ^{-1} x=\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)has

If \({ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha \) then x² = _____________

The number of solutions of the equation \({ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\frac { \pi }{ 4 } \) ____________

If \(\alpha ={ tan }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \frac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \) __________

\({ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 11 } \right) \) = ____________

If \({ tan }^{ -1 }\left( \frac { x+1 }{ x-1 } \right) +{ tan }^{ -1 }\left( \frac { x-1 }{ x } \right) ={ tan }^{ -1 }\left( -7 \right) \) then x is ___________

If \({ cos }^{ -1 }x>x>{ sin }^{ -1 }x\) then _________

The domain of cos^-1(x² - 4) is______

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

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

If x < 0, y < 0 such that xy = 1, then tan^-1(x) + tan^-1(y) =_____

The pricipal value of \({ sin }^{ -1 }\left( \frac { -1 }{ 2 } \right) \) is _________

Find the principal value of sin^-1\(\left( -\frac { 1 }{ 2 } \right) \)(in radians and degrees).

Find the period and amplitude of y = sin 7x

State the reason for cos^-1\([cos(-\frac{\pi}{6})]\neq \frac{\pi}{6}.\)

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

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

Find all values of x such that
-5\(\pi\le x \le 5\pi\) and cos x =1

Find the principal value of sin^-1(-1).

If \({ cot }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)

Evaluate \(sin\left( { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \right) \)

Prove that \(2{ tan }^{ -1 }\left( \frac { 2 }{ 3 } \right) ={ tan }^{ -1 }\left( \frac { 12 }{ 5 } \right) \)

Find the domain of sin⁻¹(2−3x²)

Find the domain of cos^-1\((\frac{2+sinx}{3})\)

Find tan(tan^-1(2019))

Find the value of tan⁻¹(−1 ) + cos^-1\((\frac{1}{2})+sin^-1(-\frac{1}{2})\)

Solve \({ sin }^{ -1 }\frac { 5 }{ x } +{ sin }^{ -1 }\frac { 12 }{ x } =\frac { \pi }{ 2 } \)

Find the number of solution of the equation tan^-1(x-1) +  tan^-1x + tan^-1(x + 1) = tan^-1(3x)

Find the domain of
g(x) = sin⁻¹x + cos⁻¹x

Find the value of
\(cot\left( { sin }^{ -1 }\frac { 3 }{ 5 } +{ sin }^{ -1 }\frac { 4 }{ 5 } \right) \)

Prove that \({ cos }^{ -1 }\left( \frac { 4 }{ 5 } \right) +{ tan }^{ -1 }\left( \frac { 3 }{ 5 } \right) ={ tan }^{ -1 }\left( \frac { 27 }{ 11 } \right) \)

Solve: cos(tan^-1x) = \(sin\left( { cot }^{ -1 }\frac { 3 }{ 4 } \right) \) 

Find the principal value of cos⁻¹\(\left( \frac { \sqrt { 3 } }{ 2 } \right) \)

Find the principal value of cosec⁻¹(−1)

Find the principal value of
sec⁻¹(−2).

Find the domain of the following functions
(i) f(x) = sin^-1(2x - 3)
(ii) f(x) = sin^-1x + cos x

Write the function \(f(x)=\tan ^{-1} \sqrt{\frac{a-x}{a+x}}-a<x<a \) in the simplest form

If \({ tan }^{ -1 }\left( \frac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right) =a\) than prove that x² = sin 2a

Prove that \({ tan }^{ -1 }\left( \frac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \frac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \frac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right)\)