Question

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

• A. $4/5$
• B. $5/2$
• C. $5/4$
• D. $3/4$

Answer 1

The correct option is D $3/4$

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

$={\cos }^2(\pi -\frac{2\pi }{5})+{\cos }^2(\pi -\frac{\pi }{5})$

$={\cos }^2\frac{2\pi }{5}+{\cos }^2\frac{\pi }{5}$

$=1-{\sin }^2\frac{2\pi }{5}+{\cos }^2\frac{\pi }{5}$

$=1+{\cos }^2\frac{\pi }{5}-{\sin }^2\frac{2\pi }{5}$

We know that ${\cos }^{2}A-{\cos }^{2}B=\cos (A+B)\cos (A-B)$

$=1+\cos \left(\frac{3\pi }{5}\right)\cos \left(\frac{\pi }{5}\right)$

$=1+\cos {108}^{\circ }\cos {36}^{\circ }$

$=1+\cos ({90}^{\circ }+{18}^{\circ })\cos {36}^{\circ }$

$=1-\sin {18}^{\circ }\cos {36}^{\circ }$

$=1-\frac{\sqrt{5}-1}{4}\times \frac{\sqrt{5}+1}{4}$

$=1-\frac{5-1}{16}$

$=1-\frac{4}{16}$

$=1-\frac{1}{4}=\frac{4-1}{4}=\frac{3}{4}$