Question

The value of ${\cos }^2\frac{3\pi }{5}+{\cos }^2\frac{4\pi }{5}$is

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

Answer 1

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

The value of ${\cos }^2\frac{3\pi }{5}+{\cos }^2\frac{4\pi }{5}$ is
$={\cos }^2{108}^{\circ }+{\cos }^2{144}^{\circ }$
$={\cos }^2(90+18{)}^{}+{\cos }^2(180-36{)}^{}$
$={\sin }^2{18}^{\circ }+{\cos }^2{36}^{\circ }$
$={\left(\frac{\sqrt{5}-1}{4}\right)}^2+{\left(\frac{\sqrt{5}+1}{4}\right)}^2$
$=\frac{2(5+1)}{16}=\frac{3}{4}$