if (a < 0) and x \(\varepsilon \) (-a, a), simplify tan^-1 \(\left( \frac { x }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } \right) \)

Write in the simplest form : \({ tan }^{ -1 }\left[ \frac { cos\quad x }{ 1+sin\quad x } \right] ,x\left[ -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right] \)

Show that :  \({ tan }^{ -1 }\frac { x }{ y } -{ tan }^{ -1 }\frac { x-y }{ x+y } =\frac { \pi }{ 4 } \)

Show that : \({ tan }^{ -1 }\frac { 2 }{ 3 } =\frac { 1 }{ 2 } { tan }^{ -1 }\frac { 12 }{ 5 } \)

Prove that sin^-1x + sin^-1y = sin^-1 \((x\sqrt { 1-{ y }^{ 2 } } +y\sqrt { 1-{ x }^{ 2 } } )if{ x }^{ 2 }-{ y }^{ 2 }\le 1\) 

Write in the simplest form \({ sin }^{ -1 }\left[ \frac { x+\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 2 } } \right] ,-\frac { 1 }{ \sqrt { 2 } } <x<\frac { 1 }{ \sqrt { 2 } } \)

Show that:
\({ tan }^{ -1 }\frac { 1 }{ 2 } +{ tan }^{ -1 }\frac { 2 }{ 11 } ={ tan }^{ -1 }\frac { 3 }{ 4 } \)

Express \(({ tan }^{ -1 }\left( \frac { cosx }{ 1-sinx } \right) ,-\frac { 3\pi }{ 2 }\) in the simplest form :

Find te principal values of the following: \({ tan }^{ -1 }\left( -\sqrt { 3 } \right) \)

Find te principal values of the following: \({ cot }^{ -1 }\left( \sqrt { 3 } \right) \)

Find te principal values of the following: \({ cosec }^{ -1 }\left( -\sqrt { 2 } \right) \)

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

Write the following functions in the simplest form:
\({ tan }^{ -1 }\left[ \frac { 3{ a }^{ 2 }x-{ x }^{ 3 } }{ { a }^{ 3 }-3{ a }^{ 2 } } \right] ,\ a>0;\ -\frac { a }{ \sqrt { 3 } } \le x\le \frac { a }{ \sqrt { 3 } } \)

Does the following trigonometric equation have any solutions? If yes, obtain the solutions (s);
\(tan^{ -1 }\left( \frac { x+1 }{ x-1 } \right) +tan^{ -1 }\left( \frac { x-1 }{ x } \right) =-tan^{ -1 }7\)

Prove that \(\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right)=\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} x^{2}\)