Question

Упростите выражение
cos²(α-π/6)+cos²(α+π/6)+sin²α

Answer 1

$cos^2(a-frac{pi}{6})+cos^2(a+frac{pi}{6})+sin^2a=\\= frac{1+cos(2a-frac{pi}{3})}{2} +frac{1+cos(2a+frac{pi}{3})}{2}+frac{1-cos2a}{2}=\\= frac{1}{2}cdot (3+underbrace {cos(2a-frac{pi}{3})+cos(2a+frac{pi}{3})}-cos2a)=\\=frac{1}{2}cdot (3+2cdot cos2acdot cos(-frac{pi}{3})-cos2a)=\\=frac{1}{2}cdot (3+2cdot frac{sqrt3}{2}cdot cos2a-cos2a)=frac{1}{2}cdot (3+(sqrt3-1)cdot cos2a)=\\=frac{3}{2}+frac{sqrt3-1}{2}cdot cos2a$