Find the principal value of \({ cot }^{ -1 }\left( -\frac { 1 }{ \sqrt { 3 } } \right) \)

3

(a)

Show that :
\({ sin }^{ -1 }(2x\sqrt { 1-{ x }^{ 2 } } )={ 2sin }^{ -1 }x,\frac { 1 }{ \sqrt { 2 } } \le x\le \frac { 1 }{ \sqrt { 2 } } \)

3

(a)

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

3

(a)

Write \({ cot }^{ -1 }\left( \frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \right) ,x>1\) in the simplest form.

3

(a)

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

3

(a)

Find the value of cos \(({ sec }^{ -1 }x+{ cosec }^{ -1 }x),\left| x \right| \ge 1\)

3

(a)

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

3

(a)

Show that :
\({ sin }^{ -1 }\frac { 12 }{ 13 } +cos^{ -1 }\frac { 4 }{ 5 } =tan^{ -1 }\frac { 63 }{ 16 } =\pi \)

3

(a)

Simplify :
\({ tan }^{ -1 }\left( \frac { acosx-bsinx }{ bcos+asinx } \right) \), if \(\frac { a }{ b } tanx>-1\)

3

(a)

Find te principal values of the following:
\({ cos }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right) \)

3

(a)

Find te principal values of the following: cosec^-1(2)

3

(a)

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

3

(a)

Find te principal values of the following: tan^-1(-1)

3

(a)

Find te principal values of the following: \({ sec }^{ -1 }\left( \frac { 2 }{ \sqrt { 3 } } \right) \)

3

(a)

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

3

(a)