Question

Find the value of ${\cos }^2\frac{3\pi }{5}+{\cos }^2\frac{4\pi }{5}$

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

Answer 1

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

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

$=\frac{1+\cos \frac{6\pi }{5}}{2}+\frac{1+\cos \frac{8\pi }{5}}{2}$

$=\frac{1}{2}[2+\cos \frac{6\pi }{5}+\cos \frac{8\pi }{5}]$

$=\frac{1}{2}[2+\cos (180°+36°)+\cos (360°-72°)]$

$=\frac{1}{2}[2-\cos 36°+\cos (90°-18°)]$

$=\frac{1}{2}[2-\cos 36°+\sin 18°]$

$=\frac{1}{2}[2-\frac{\sqrt{5}+1}{4}+\frac{-1+\sqrt{5}}{4}]$

$=\frac{1}{2}[2-\frac{\sqrt{5}}{4}-\frac{1}{4}-\frac{1}{4}+\frac{\sqrt{5}}{4}]$

$=\frac{1}{2}[2-\frac{1}{2}]$

$=\frac{1}{2}\times \frac{3}{2}$

$=\frac{3}{4}$

Hence, the answer is $\frac{3}{4}.$