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 $