Question

If $f\left(x\right)={\sin }^{-1}\left(\sin x\right)+{\cos }^{-1}\left(\sin x\right)\text{and}\varphi \left(x\right)=f\left(f\right(f\left(x\right)\left)\right)$ then ${\varphi }^{\prime }\left(x\right)$

• A. 1
• B. $\sin x$
• C. 0
• D. none of these

Answer 1

The correct option is C 0

$f\left(x\right)={\sin }^{-1}\left(\sin x\right)+{\cos }^{-1}\left(\sin x\right)=\frac{\pi }{2}[\because {\sin }^{-1}\theta +{\cos }^{-1}\theta =\frac{\pi }{2}]$

$f\left(f\right(x\left)\right)=f\left(\frac{\pi }{2}\right)=\frac{\pi }{2}$

$\varphi \left(x\right)=f\left(f\right(f\left(x\right)\left)\right)=f\left(\frac{\pi }{2}\right)=\frac{\pi }{2}$

$\varphi \left(x\right)=\frac{\pi }{2}$

${\varphi }^{\prime }\left(x\right)=0$