Question

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

Answer 1

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

Ответ : $\frac{2\pi n }{3},n\in Z$

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.