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

Evaluate \(cos\left[ { sin }^{ -1 }\frac { 3 }{ 5 } +{ sin }^{ -1 }\frac { 5 }{ 13 } \right] \)

Prove that \({ tan }^{ -1 }\left( \frac { m }{ n } \right) -{ tan }^{ -1 }\left( \frac { m-n }{ m+n } \right) =\frac { \pi }{ 4 } \)

Solve \({ tan }^{ -1 }\left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) +{ cot }^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 2x } \right) =\frac { \pi }{ 3 } ,x>0\)

If \(sin\left( { sin }^{ -1 }\frac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1\) then find the value ofx.

Prove that \({ tan }^{ -1 }\sqrt { x } =\frac { 1 }{ 2 } { cos }^{ -1 }={ \frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 1-x }{ 1+x } \right) ,x\in \left| 0,1 \right| }\)

Evaluate \(cos\left[ { cos }^{ -1 }\left( \frac { -\sqrt { 3 } }{ 2 } +\frac { \pi }{ 6 } \right) \right] \)

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

Solve that \(\sin ^{-1}\left(2 x \sqrt{1-x^{-2}}\right)=2 \sin ^{-1} x, \frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}}\)

Evaluate \(\sin ^{-1}\left(\sin \left(-600^{\circ}\right)\right)\)

Find the value of \(\cot \left(\tan ^{-1} a+\cot ^{-1} a\right)\)

Prove that \(\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}, x \in[-1,1]\)

Find value of \(\tan \left(\cos ^{-1} x\right)\) and hence evaluate \(\tan \left(\cos ^{-1} \frac{8}{17}\right)\)

Prove that \(\tan \left(\cot ^{-1} x\right)=\cot \left(\tan ^{-1} x\right)\). State with reason whether the equality is valid for all values of x.