Question

cos^2(x/2) - sin^2(x/2) = cos2x

Answer 1

$Cos^{2}frac{x}{2}-Sin^{2}frac{x}{2}=Cos2x\\Cos(2*frac{x}{2})=Cos2x\\Cos2x-Cosx=0\\-2Sinfrac{2x-x}{2}Sinfrac{2x+x}{2}=0\\Sinfrac{x}{2}Sinfrac{3x}{2}=0\\1)Sinfrac{x}{2}=0\\frac{x}{2}=pi n,nin Z\\x=2pi n,nin Z\\2)Sinfrac{3x}{2}=0\\frac{3x}{2}=pi n,nin Z\\x=frac{2pi n }{3},nin Z$

Ответ : $frac{2pi n }{3},nin Z$

Answer 2

cos^2(x/2) - sin^2(x/2) = cos 2x
cos x = cos 2x
cos 2x - cos x = 0
-2•sin(3x/2)•sin(x/2) = 0

sin(3x/2) = 0
3x/2 = πn, n є Z
x = 2πn/3, n є Z

sin(x/2) = 0
x/2 = πk, k є Z
x = 2πk, k є Z

Ответ: x = 2πn/3, n є Z.