Write in the simplest form: \(({ tan }^{ -1 }\left[ \frac { \sqrt { 1+sin\quad x } +{ \sqrt { 1-sin\quad x } }\quad }{ \sqrt { 1+sin\quad x } +{ \sqrt { 1-sin\quad x } } } \right] ,0<x<\frac { \pi }{ 2 } \)

4

(a)

Solve for X, 2tan^-1(sin x) = tan^-1(2sec x), \(x\neq \frac { \pi }{ 2 } \)

4

(a)

Solve the following equations:
\(\tan ^{-1}\left(\frac{x+1}{x-1}\right)+\tan ^{-1}\left(\frac{x-1}{x}\right)=\tan ^{-1}(-7)\)

4

(a)

If \({ a }_{ 1 },{ a }_{ 2 },{ a }_{ 3 }.........{ a }_{ n }\) is and arithmetic progressive with common difference d, then evaluate,
 \(tan\left[ { tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +{ tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +{ tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ 3 }{ a }_{ 4 } } \right) +.....{ tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ n-1 }{ a }_{ n } } \right) \right] \)

4

(a)

If \(y=\cot ^{ -1 }{ (\sqrt { \cos { x } } ) } -\tan ^{ -1 }{ (\sqrt { \cos { x } } ) } ,\)  then prove the  that \(\sin { y } =\tan ^{ 2 }{ \left( \frac { x }{ 2 } \right) } .\)

4

(a)

Solve the equation for x : sin^-1 x + sin^-1(1-x) = cos^-1x.

4

(a)

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

4

(a)

Show that \(\sin { \left[ \cot ^{ -1 }{ (\cos { (\tan ^{ -1 }{ x } )) } } \right] } =\frac { \sqrt { { x }^{ 2 }+1 } }{ \sqrt { { x }^{ 2 }+2 } } .\)

4

(a)

Prove that \(\tan ^{ -1 }{ \left\{ \frac { \sqrt { 1+\cos { x } } +\sqrt { 1-\cos { x } } }{ \sqrt { 1+\cos { x } } -\sqrt { 1-\cos { x } } } \right\} } =\frac { \pi }{ 4 } +\frac { x }{ 2 } ,\quad if0

4

(a)

Show that \({ cot }^{ -1 }\left( \frac { \sqrt { 1+sin\quad x } +\sqrt { 1-sin\quad x } }{ \sqrt { 1+sin\quad x } -\sqrt { 1-sin\quad x } } \right) =\frac { x }{ 2 } ,x\in \left( 0,\frac { \pi }{ 4 } \right) \)

4

(a)

Prove that \({ cos }^{ -1 }(x)+{ cos }^{ -1 }\left( \frac { x }{ 2 } +\frac { \sqrt { 3-{ 3x }^{ 2 } } }{ 2 } \right) =\frac { \pi }{ 3 } \)

4

(a)

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

4

(a)

If \(\sin ^{ -1 }{ x } +\sin ^{ -1 }{ y } +\sin ^{ -1 }{ z } =\pi ,\) then prove that \(x\sqrt { 1-{ x }^{ 2 } } +y\sqrt { 1-{ y }^{ 2 } } +z\sqrt { 1-{ z }^{ 2 } } =2xyz.\)

4

(a)

Prove that 2\({ tan }^{ -1 }\left( \frac { 1 }{ 5 } \right) +{ sec }^{ -1 }\left( \frac { 5\sqrt { 2 } }{ 7 } \right) +2{ tan }^{ -1 }\left( \frac { 1 }{ 8 } \right) =\frac { \pi }{ 4 } \)

4

(a)

Prove the following: \(\cos { \left[ \tan ^{ -1 }{ \left\{ \sin { (\cot ^{ -1 }{ x } ) } \right\} } \right] } =\sqrt { \frac { 1+{ x }^{ 2 } }{ 2+{ x }^{ 2 } } } \)

4

(a)

Solve the equation: \(\tan ^{ -1 }{ \sqrt { { x }^{ 2 }+x } } +\sin ^{ -1 }{ \sqrt { { x }^{ 2 }+x+1 } } =\frac { \pi }{ 2 } .\)

4

(a)

Prove that : \({ tan }^{ -1 }\left[ \frac { \sqrt { 1+x } -\sqrt { 1-x } }{ \sqrt { 1+x } +\sqrt { 1-x } } \right] =\frac { \pi }{ 4 } -\frac { 1 }{ 2 } { cos }^{ -1 }x,-\frac { 1 }{ \sqrt { 2 } } \le x\le 1.\)

4

(a)

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

4

(a)