KVPY Sample Paper KVPY Stream-SX Model Paper-27

If \[x\in \left( \frac{3\pi }{2},2\pi \right)\] then value of the expression \[{{\sin }^{-\,1}}\,(\cos \,({{\cos }^{-\,1}}(\cos x)+{{\sin }^{-\,1}}\,(\sin x)))\] equals-

A) \[-\frac{\pi }{2}\]

B) \[\frac{\pi }{2}\]

C) 0

D) None of these

Correct Answer: B

Solution :

\[\because \,\,{{\cos }^{-\,1}}\cos x={{\cos }^{-\,1}}\cos (2\pi -x)\] For \[x\in \left( \frac{3\pi }{2},2\pi \right)=2\pi -x\]

\[{{\sin }^{-1}}\sin x=x-2\pi \]

\[\therefore \,\,{{\sin }^{-\,1}}(\cos \,({{\cos }^{-\,1}}(\cos x)+{{\sin }^{-\,1}}(\sin x)))\]

\[={{\sin }^{-1}}(\cos 0)={{\sin }^{-1}}(1)=\frac{\pi }{2}\]