The value of sin^-1 (cos x), \(0\le x\le\pi\) is

If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is

If \(\sin ^{-1} x+\cot ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{2}\), then x is equal to

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

If \(\alpha ={ tan }^{ -1 }\left( tan\frac { 5\pi }{ 4 } \right) \) and \(\beta ={ tan }^{ -1 }\left( -tan\frac { 2\pi }{ 3 } \right) \) then ___________

The number of real solutions of the equation \(\sqrt { 1+cos2x } ={ 2sin }^{ -1 }\left( sinx \right) ,-\pi  is ___________

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

\(sin\left\{ 2{ cos }^{ -1 }\left( \frac { -3 }{ 5 } \right) \right\} =\) __________

\(cot\left( \frac { \pi }{ 4 } -{ cot }^{ -1 }3 \right) \)

If tan^-1(cot \(\theta\)) = 2\(\theta\), then\(\theta\) = _____________

The value of sin 2(tan^-1 0.75) is ___________

If \(\theta ={ sin }^{ -1 }\left( sin(-{ 60 }^{ 0 }) \right) \) then one of the possible values of \(\theta\) 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) =_____

\({ tan }^{ -1 }\left( tan\cfrac { 9\pi }{ 8 } \right) \)

Find the value of sin^-1\(\left( sin\frac { 5\pi }{ 9 } cos\frac { \pi }{ 9 } +cos\frac { 5\pi }{ 9 } sin\frac { \pi }{ 9 } \right) \).

Show that cot⁻¹\(\left( \frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \right) ={ sec }^{ -1 }x,|x|>1\)

Prove that tan \(\left( { sin }^{ -1 }x \right) =\frac { x }{ \sqrt { 1-{ x }^{ 2 } } } for|x|<1\)

Find the period and amplitude of y = 4sin(−2x)

Find the value of \({ sin }^{ -1 }\left( sin\left( \frac { 5\pi }{ 4 } \right) \right) \)​

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

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

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

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

Evaluate \(sin\left( { cos }^{ -1 }\left( \frac { 1 }{ 2 } \right) \right) \)

Find the domain of the following
 \(f\left( x \right) { =sin }^{ -1 }\left( \frac { { x }^{ 2 }+1 }{ 2x } \right) \)

For what value of x, the inequality \(\frac { \pi }{ 2 } <{ cos }^{ -1 }(3x-1)<\pi \) holds?

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 x > cos^-1x

Prove that \({ tan }^{ -1 }x+{ tan }^{ -1 }\frac { 2x }{ 1-{ x }^{ 2 } } ={ tan }^{ -1 }\frac { 3x-{ x }^{ 3 } }{ 1-{ 3x }^{ 2 } } ,|x|<\frac { 1 }{ \sqrt { 3 } } \)

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

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

Find the real solutions of the equation
\({ tan }^{ -1 }\sqrt { x(x+1) } +{ sin }^{ -1 }\sqrt { { x }^{ 2 }+x+1 } =\frac { \pi }{ 2 } \)

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 cos^-1 \((-\frac{1}{\sqrt2})\)

If a₁, a₂, a₃, ... an is an arithmetic progression with common difference d, prove that tan\( \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } } \)

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

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