Ta có $\cos\alpha=-\cos(180^o-\alpha)\Rightarrow \cos^2\alpha=\cos^2(180^o-\alpha)$
$\cos^215^o+\cos^245^o+\cos^2135^o+\cos^2165^o$
$=\cos^215^o+\cos^245^o+\cos^2(180^o-135^o)+\cos^2(180^o-165^o)$
$=\cos^215^o+\cos^245^o+\cos^245^o+\cos^215^o$
$=2\cos^215^o+2\cos^245^o$
$=S$
Ta có $\cos2\alpha=\cos^2\alpha-\sin^2\alpha=\cos^2\alpha-(1-\cos^2\alpha)=2\cos^2\alpha-1$
Do đó:
$S-2=2\cos^215^o-1+2\cos^245^o-1$
$=\cos2.15^o-\cos2.45^o$
$=\cos30^o-\cos90^o$
$=\dfrac{\sqrt3}{2}$
$\Rightarrow S=\dfrac{\sqrt3}{2}+2=\dfrac{4+\sqrt3}{2}$
