Question

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

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

Answer 1

The correct option is D $\sqrt{5}/3$

Let $co{t}^{-1}\frac{1}{2}$ be $\theta$.
Then, $sin\theta =\frac{2}{\sqrt{5}}$
$\Rightarrow$ R.H.S. $=\frac{2}{\sqrt{5}}$
Now, let $co{s}^{-1}x$ be $\alpha .$
$\Rightarrow x=cos\alpha$ and $tan\alpha =\frac{\sqrt{1-x^2}}{x}=$ L.H.S.
Equating L.H.S. and R.H.S., we have $\frac{\sqrt{1-x^2}}{x}=\frac{2}{\sqrt{5}}$
$\Rightarrow \sqrt{5-5x^2}=2x$ which on squaring gives $5-5x^2=4x^2$
Or, $x=\frac{\sqrt{5}}{3}$