Mathematics - Functions Question with Solution | TestHub

$f(x)=2{\sin }^2\theta +4\cos (x+\theta )\sin x\sin \theta +\cos (2x+2\theta )$

$$\begin{array}{rl}& =2{\sin }^2\theta +\cos (2x+2\theta )+2\cos (x+\theta )\cos (x-\theta )-2{\cos }^2(x+\theta )\\ & =2{\sin }^2\theta +2{\cos }^2(x+\theta )-1+2{\cos }^{2}x-2{\sin }^2\theta -2{\cos }^2(x+\theta )\\ & =\cos 2x\\ & \therefore f^2(x)+f^2\left(\frac{\pi }{4}-x\right)={\cos }^{2}2x+{\sin }^{2}2x=1\end{array}$$