Question

$\sin ({\cot }^{-1}(\tan {\cos }^{-1}x)$ is equal to

• A. $x$
• B. $\sqrt{1-x^2}$
• C. $\frac{1}{x}$
• D. none of these

Answer 1

Answer: (A)

Put $co{s}^{-1}x=\theta$
$\therefore x=cos\theta$
$\therefore tan\theta =\frac{\sqrt{1-x^2}}{x}$

$\therefore tan\left(co{s}^{-1}x\right)=tan\theta =\frac{\sqrt{1-x^2}}{x}$
$\therefore co{t}^{-1}\left(tan\left(co{s}^{-1}\right)\right)=co{t}^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)=\varphi$ (say)

$\therefore cot\varphi =\frac{\sqrt{1-x^2}}{x}$
$\therefore sin\varphi =\frac{x}{1}$
$\therefore sin(co{t}^{-1}\left(tan\left(co{s}^{-1}x\right)\right)$
$=sin\varphi =x$