Question

The value of ${\sin }^2\frac{\pi }{16}+{\sin }^2\frac{3\pi }{16}+{\sin }^2\frac{5\pi }{16}+{\sin }^2\frac{7\pi }{16}$
is

• A. $1$
• B. $2$
• C. $0$
• D. $-2$

Answer 1

The correct option is B $2$

Given,
${\sin }^2\frac{\pi }{16}+{\sin }^2\frac{3\pi }{16}+{\sin }^2\frac{5\pi }{16}+{\sin }^2\frac{7\pi }{16}$
We know that
$\frac{\pi }{16}+\frac{7\pi }{16}=\frac{\pi }{2};\frac{3\pi }{16}+\frac{5\pi }{16}=\frac{\pi }{2}$

Now,
$({\sin }^2\frac{\pi }{16}+{\sin }^2\frac{7\pi }{16})+({\sin }^2\frac{3\pi }{16}+{\sin }^2\frac{5\pi }{16})=({\sin }^2\frac{\pi }{16}+{\cos }^2\frac{\pi }{16})+({\sin }^2\frac{3\pi }{16}+{\cos }^2\frac{3\pi }{16})=1+1=2$