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

5

(a)

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

5

(a)

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

5

(a)

Solve for \(x:\tan ^{ -1 }{ 3x } +\tan ^{ -1 }{ 2x } =\frac { \pi }{ 4 } .\)

5

(a)

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

5

(a)

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

5

(a)