Question

The value of ${\cos }^4\left(\frac{\pi }{8}\right)+{\cos }^4\left(\frac{3\pi }{8}\right)+{\cos }^4\left(\frac{5\pi }{8}\right)+{\cos }^4\left(\frac{7\pi }{8}\right)$ is equal to

• A. $\frac{1}{2}$
• B. $\frac{3}{4}$
• C. $\frac{3}{2}$
• D. $1$

Answer 1

The correct option is C $\frac{3}{2}$

${\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4\frac{5\pi }{8}+{\cos }^4\frac{7\pi }{8}={\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4(\pi -\frac{3\pi }{8})+{\cos }^4(\pi -\frac{\pi }{8})=2({\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8})=2({\cos }^4\frac{\pi }{8}+{\cos }^4(\frac{\pi }{2}-\frac{\pi }{8}))=2({\cos }^4\frac{\pi }{8}+{\sin }^4\frac{\pi }{8})=2(1-2{\sin }^2\frac{\pi }{8}{\cos }^2\frac{\pi }{8})=2(1-\frac{1}{2}{\sin }^2\frac{\pi }{4})=2(1-\frac{1}{2}.\frac{1}{2})=\frac{3}{2}$