Question

${\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2\frac{5\pi }{8}+{\sin }^2\frac{7\pi }{8}$ is equal to

• A. $1$
• B. $\frac{3}{2}$
• C. 2
• D. $\frac{5}{2}$

Answer 1

Answer: (C)

${\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2\frac{5\pi }{8}+{\sin }^2\frac{7\pi }{8}$
${\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2\left(\pi -\frac{3\pi }{8}\right)+{\sin }^2\left(\pi -\frac{\pi }{8}\right)$
${\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2\frac{\pi }{8}$
$=2\left[{\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}\right]$
$=2\left[{\sin }^2\frac{\pi }{8}+{\sin }^2\left(\frac{\pi }{2}-\frac{\pi }{8}\right)\right]$
$=2\left[{\sin }^2\frac{\pi }{8}+{\cos }^2\frac{\pi }{8}\right]$
$=2.1=2$