Question

Prove that
${\cos }^{2}A+{\cos }^2(A+\frac{\pi }{3})+co{s}^2(A-\frac{\pi }{3})=\frac{3}{2}$

Answer 1

${\cos }^{2}A+{\cos }^2(A+\frac{\pi }{3})+{\cos }^2(A-\frac{\pi }{3})$

$={\cos }^{2}A+2\left({\cos }^{2}A{\cos }^2\right(\frac{\pi }{3})+{\sin }^{2}A{\sin }^2(\frac{\pi }{3})$

$={\cos }^{2}A+\frac{{\cos }^{2}A}{2}+\frac{3}{2}{\sin }^{2}A$

$=\frac{3}{2}({\cos }^{2}A+{\sin }^{2}A)$

$=\frac{3}{2}$