Question

Find the value of ${cos}^2\frac{\pi }{10}+$ ${cos}^2\frac{2\pi }{5}+$ ${cos}^2\frac{3\pi }{5}+$ ${cos}^2\frac{9\pi }{10}$.

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

Answer 1

The correct option is D $2$

$\cos \left(\frac{9\pi }{10}\right)=\cos (\pi -\frac{\pi }{10})=-\cos \left(\frac{\pi }{10}\right)$
$\cos \left(\frac{2\pi }{5}\right)=\cos (\pi -\frac{3\pi }{5})=-\cos \left(\frac{3\pi }{5}\right)$
Hence, the equation can be written as
$2({\cos }^2\frac{\pi }{10}+{\cos }^2\frac{3\pi }{5})$
$\cos \left(\frac{3\pi }{5}\right)=\sin (\frac{\pi }{2}-\frac{3\pi }{5})=-\sin \left(\frac{\pi }{10}\right)$
Thus the equation becomes
$2({\cos }^2\frac{\pi }{10}+{\sin }^2\frac{\pi }{10})$
$=2$