Question

Evaluate

$\begin{array}{cc}& {\cos }^4\frac{\pi }{16}+{\sin }^4\frac{\pi }{16}-{\sin }^2\frac{\pi }{16}{\cos }^2\frac{\pi }{16}+{\cos }^4\frac{3\pi }{16}+{\sin }^4\frac{3\pi }{16}-{\sin }^2\frac{3\pi }{16}{\cos }^2\frac{3\pi }{16}\\ & \end{array}$

Answer 1

$\Rightarrow \cos {}^2\frac{\pi }{16}+\sin {}^4\frac{\pi }{16}-\sin {}^2\frac{\pi }{16}\cos {}^2\frac{\pi }{16}+\cos {}^4\frac{3\pi }{16}+\sin {}^4\frac{3\pi }{16}-\sin {}^2\frac{3\pi }{16}\cos {}^2\frac{3\pi }{16}\Rightarrow [{(\cos {}^2\frac{\pi }{16}+\sin {}^2\frac{\pi }{16})}^2+\sin {}^2\frac{\pi }{16}\cos {}^2\frac{\pi }{16}]+[{(\cos {}^2\frac{3\pi }{16}+\sin {}^2\frac{3\pi }{16})}^2+\cos {}^2\frac{3\pi }{16}\sin {}^2\frac{3\pi }{16}]\Rightarrow 2+\frac{1}{4}\left(4\right)\sin {}^2\frac{\pi }{16}\cos {}^2\frac{\pi }{16}+\frac{1}{4}\left(4\right)\sin {}^2\frac{3\pi }{16}\cos {}^2\frac{3\pi }{16}\Rightarrow 2+2(\sin {}^2\frac{\pi }{8}+\sin {}^2\frac{3\pi }{8})\Rightarrow 2+2(\sin {}^2\frac{\pi }{8}+\sin {}^2(\frac{\pi }{2}-\frac{\pi }{8}))\Rightarrow 2+2(\sin {}^2\frac{\pi }{8}+\cos {}^2\frac{\pi }{8})\Rightarrow 2+2\Rightarrow 4$