Why is

\(cos^2 x = \frac{1}{2} + \frac{cos(2x)}{2}\)

?

 

From the identity
cos(x+y)=cosxcosy-sinxsiny
Now if x=y
cos 2x=cos^2 x - sin ^2 x
cos 2x=cos ^2 x- (1-cos^2 x)
cos 2x=2 cos^2 x - 1
cos ^2 x= (1+cos 2x)/2