Question

${\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4\frac{5\pi }{8}+{\cos }^4\frac{7\pi }{8}$ is equal to

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

Answer 1

Answer: (A)

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