Solve: \({ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 } \)

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

Simplify \({ sin }^{ -1 }\left( \frac { sinx+cosx }{ \sqrt { 2 } } \right) ,\frac { \pi }{ 4 }\) 

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

Solve that \(\sin ^{-1} \frac{3}{5}-\sin ^{-1} \frac{8}{17}=\cos ^{-1} \frac{84}{85}\)

Find the value of \(\sin \left[2 \cot ^{-1}\left(\frac{-5}{12}\right)\right]\)

Find the value of \(\tan \frac{1}{2}\left[\sin ^{-1} \frac{2 x}{1+x^{2}}+\cos ^{-1} \frac{1-y^{2}}{1+y^{2}}\right]\), \(|x|<1, y>0,|x y|<1\)

Write the function in simplest form \(\tan ^{-1} \frac{\sqrt{1+x^{2}}-1}{x}, x \neq 0\)

Prove that \(2 \sin ^{-1} \frac{3}{5}-\tan ^{-1} \frac{17}{31}=\frac{\pi}{4}\)

Prove that \(\cot ^{-1} 7+\cot ^{-1} 8+\cot ^{-1} 18=\cot ^{-1} 3\)

Find the values of x which satisfy the equation \(\sin ^{-1} x^{-2}+\sin ^{-1}(1+x)=\cos ^{-1} x .\)

Show that \(2 \tan ^{-1}\left\{\tan \frac{\alpha}{2} \cdot \tan \left(\frac{\pi}{4}-\frac{\beta}{2}\right)\right\}=\tan ^{-1} \frac{\sin \alpha \cos \beta}{\cos \alpha+\sin \beta}\)

Which is greater, tan 1 or tan^-11?