Question

$\mathrm{If}y={\sin }^{-1}\left(\cos x\right),\mathrm{where}x\in \left(0,2\pi \right),\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\mathrm{dy}}{\mathrm{dx}}\mathrm{is}$

• A. $-1,if0<x<\pi$
• B. $1,if\pi <x<2\pi$
• C. $1,if0<x<\pi$
• D. $-1,if\pi <x<2\pi$

Answer 1

Answer: (A), (B)

The correct options are
A $-1,if0<x<\pi$
B $1,if\pi <x<2\pi$

$\mathrm{We}\mathrm{have},{\sin }^{-1}\left(\cos x\right)=\frac{\pi }{2}-{\cos }^{-1}\left(\cos x\right)=(\begin{array}{c}\frac{\pi }{2}-x,\mathrm{if}0<x<\pi \\ \frac{\pi }{2}-\left(2\pi -x\right),\mathrm{if}\pi <x<2\pi \end{array}=(\begin{array}{c}\frac{\pi }{2}-x,\mathrm{if}0<x<\pi \\ x-\frac{3\pi }{2},\mathrm{if}\pi <x<2\pi \end{array}\mathrm{Clearly},\mathrm{it}\mathrm{is}\mathrm{not}\mathrm{differentiable}\mathrm{at}x=\pi .\mathrm{Therefore},\frac{d}{\mathrm{dx}}\left({\sin }^{-1}\left(\cos x\right)\right\}=(\begin{array}{c}-1,\mathrm{if}0<x<\pi \\ 1,\mathrm{if}\pi <x<2\pi \end{array}$