$f(x)={{\sin }^{2}}(f'(x))$ $\Rightarrow f'(x)=f''(x)\times 2\sin (f'(x))\times \cos (f'(x))$ $=f''(x)\times \sin (2f'(x))\xrightarrow{x=0}f'(0)=f''(0)\times \sin (2f'(0))$ $\xrightarrow{f'(0)=\frac{\pi }{4}}f'(0)=f''(0)\times \sin \left( 2\left( \frac{\pi }{4} \right) \right)$ $\Rightarrow f''(0)=f'(0)=\frac{\pi }{4}$