Question

The value of the expression ${\sin }^4\frac{\pi }{8}+{\sin }^4\frac{3\pi }{8}+{\sin }^4\frac{5\pi }{8}+{\sin }^4\frac{7\pi }{8}$ is

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

Answer 1

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

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