Question

Considering only the principal values, if

$\tan \left( \cos ^ { - 1 } x \right) = \sin \left[ \cot ^ { - 1 } \left( \frac { 1 } { 2 } \right) \right]$, then x is equal to

• A. $\frac { 1 } { \sqrt { 5 } }$
• B. $\frac { 2 } { \sqrt { 5 } }$
• C. $\frac { 3 } { \sqrt { 5 } }$
• D. $\frac { \sqrt { 5 } } { 3 }$

Answer 1

Answer: (D)

$\frac { \sqrt { 5 } } { 3 }$

Answer 2

Put $\cot ^ { - 1 } \left( \frac { 1 } { 2 } \right) = \theta \Rightarrow \cot \theta = \frac { 1 } { 2 }$

∴ $\cos ^ { - 1 } x = \phi$ then $x = \cos \phi$

Also $\tan \phi = \frac { 2 } { \sqrt { 5 } } , \therefore x = \cos \phi = \frac { \sqrt { 5 } } { 3 }$