Considering only the principal values, if ( ^ - 1 x) = [ ^ - 1 ( … - Vidyadip

MCQ

Considering only the principal values, if $\tan ({\cos ^{ - 1}}x)$ $ = \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
$\frac{3}{{\sqrt 5 }}$
✓
$\frac{{\sqrt 5 }}{3}$

✓

Answer

Correct option: D.

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

d
(d) Put ${\cot ^{ - 1}}\,\left( {\frac{1}{2}} \right) = \theta \,\, \Rightarrow \cot \theta = \frac{1}{2}$
$\therefore$ $\sin \theta = \frac{2}{{\sqrt 5 }}.$ Put ${\cos ^{ - 1}}x = \phi $, $\therefore$ $x = \cos \phi $
Also, $\tan \phi = \frac{2}{{\sqrt 5 }},\,\,\,\,\,\therefore \,x = \cos \phi = \frac{{\sqrt 5 }}{3}$.

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