Question

Evaluate:
${\cos }^2{20}^o+{\cos }^2{30}^o+{\cos }^2{40}^o+{\cos }^2{50}^o+{\cos }^2{70}^o$.

Answer 1

${\cos }^2{20}^o+{\cos }^2{30}^o+{\cos }^2{40}^o+{\cos }^2{50}^o+{\cos }^2{70}^o$

$={\cos }^2(90-{70}^o)+{\cos }^{2}30+{\cos }^2(90-50)+{\cos }^{2}50+{\cos }^{2}70$

$={\sin }^{2}70+{\cos }^{2}30+{\sin }^{2}50+{\cos }^{2}50+{\cos }^{2}70$

$=({\sin }^{2}70+{\cos }^{2}70)+{\left(\frac{\sqrt{3}}{2}\right)}^2+({\sin }^{2}50+{\cos }^{2}50)$

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

$=2+\frac{3}{4}=\frac{8+3}{4}=\frac{11}{4}$.