Question

If $y=\sin [{\cos }^{-1}\{\sin ({\cos }^{-1}x)\}],$ $\frac{dy}{dx}$ at $x=\frac{1}{2}$ is equal to

• A. $0$
• B. $-1$
• C. $\frac{2}{\sqrt{3}}$
• D. $\frac{1}{\sqrt{3}}$
• E. $1$

Answer 1

Answer: (E)

Given, $y=\sin [{\cos }^{-1}\{\sin ({\cos }^{-1}x)\}]$
$\Rightarrow$ $y=som\left[{\cos }^{-1}\left\{\sin \left(\frac{\pi }{2}-{\sin }^{-1}x\right)\right\}\right]$
$=\sin [{\cos }^{-1}(\cos {\sin }^{-1}x)]$
$=\sin ({\sin }^{-1}x)$
$\Rightarrow$ $y=x$
On differentiating w.r.t. $x,$
we get $\frac{dy}{dx}=1$