Solve cos2x=cos^2x-1/2

Hello:

by identity : cos (x+x) = cosx cosx-sinxsinx = cos² (x) - sin² (x)

but : cos² (x) + sin² (x) = 1

sin² (x) = 1-cos² (x)

cos (2x) = cos² (x) - (1-cos² (x))

cos (2x) = 2cos² (x) - 1

Solve cos (2x) = cos² (x) - 1/2

2cos² (x) - 1 = cos² (x) - 1/2

cos² (x) = 1/2

cos (x) = 1/√2 or cos (x) = - 1/√2

1) cos (x) = 1/√2 : in R x = π/4 + 2kπ or x = - π/4 + 2kπ

2) cos (x) = - 1/√2 in R x=5π/4 + 2kπ or x = - 5π/4 + 2kπ k in Z