Question

The value of ${\cos }^2\frac{\pi }{8}+{\cos }^2\frac{3\pi }{8}+{\cos }^2\frac{5\pi }{8}+{\cos }^2\frac{7\pi }{8}$ is

Answer 1

The given expression
$\frac{1}{2}(2{\cos }^2\frac{\pi }{8}+2{\cos }^2\frac{3\pi }{8}+2{\cos }^2\frac{5\pi }{8}+2{\cos }^2\frac{7\pi }{8})$
$=\frac{1}{2}(1+\cos \frac{\pi }{4}+1+\cos \frac{3\pi }{4}+1+\cos \frac{5\pi }{4}+1+\cos \frac{7\pi }{4})[\because 2{\cos }^2\theta =1+\cos 2\theta ]$
$=\frac{1}{2}[4+\cos \frac{\pi }{4}+\cos (\pi -\frac{\pi }{4})+\cos (\pi +\frac{\pi }{4})+\cos (2\pi -\frac{\pi }{4})]$
$=\frac{1}{2}(4+\cos \frac{\pi }{4}-\cos \frac{\pi }{4}-\cos \frac{\pi }{4}+\cos \frac{\pi }{4})=\frac{1}{2}\times 4=2$