Question

Find the value of ${\cos }^2\frac{\pi }{16}+{\cos }^2\frac{3\pi }{16}+{\cos }^2\frac{5\pi }{16}+{\cos }^2\frac{7\pi }{16}$

Answer 1

We have, ${\cos }^2\frac{\pi }{16}+{\cos }^2\frac{3\pi }{16}+{\cos }^2(\frac{\pi }{2}-\frac{3\pi }{16})+{\cos }^2(\frac{\pi }{2}-\frac{\pi }{16})$

$={\cos }^2\frac{\pi }{16}+{\cos }^2\frac{3\pi }{16}+{\sin }^2\frac{3\pi }{16}+{\sin }^2\frac{\pi }{16}$

$=({\cos }^2\frac{\pi }{16}+{\sin }^2\frac{\pi }{16})+({\cos }^2\frac{3\pi }{16}+{\sin }^2\frac{3\pi }{16})$

$=1+1=2$
Ans: 2