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