Question

If $\tan \left(\cos ^{-1} x\right)=\sin \left(\cot ^{-1} \frac{1}{2}\right)$ then $x=$

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

Answer 1

Answer: (C)

$\tan \left(\tan ^{-1} \frac{\sqrt{1-x^{2}}}{x}\right)=\sin \left(\sin ^{-1} \frac{2}{\sqrt{5}}\right) $
$\Rightarrow \frac{\sqrt{1-x^{2}}}{x}=\frac{2}{\sqrt{5}} $
$\Rightarrow \frac{1-x^{2}}{x}=\frac{4}{5} x ^{2}=5 / 9$
$x =\pm \frac{\sqrt{5}}{3} $
But $x =-\frac{\sqrt{5}}{3}$ does not satisfy the given equation
$ \therefore x =\frac{\sqrt{5}}{3} $