Question

${\cos }^2\frac{3\pi }{5}+{\cos }^2\frac{4\pi }{5}$ is equal to?

• A. $\frac{4}{5}$
• B. $\frac{5}{2}$
• C. $\frac{5}{4}$
• D. $\frac{3}{4}$

Answer 1

The correct option is D $\frac{3}{4}$

${\cos }^2\frac{3\pi }{5}+{\cos }^2\frac{4\pi }{5}$

$={\cos }^2\left(108^0\right)+{\cos }^2\left({144}^0\right)$

$={\cos }^2({90}^0+{18}^0)+{\cos }^2\left({144}^0\right)$

$={\sin }^2{18}^0+{\sin }^2{54}^0$

${\sin 18}^0=\frac{\sqrt{5}-1}{4}{\sin 54}^0=\frac{\sqrt{5}+1}{4}$

$={\left(\frac{\sqrt{5}-1}{4}\right)}^2+{\left(\frac{\sqrt{5}+1}{4}\right)}^2$

$=3/4$