2cos^(2)(x/2)

Add 0 in the form of (-1 + 1):

2cos^(2)(x/2) - 1 + 1

Use double angle formula with the first two terms.
We know cos(2x) = 2cos^(2)(x) - 1

cos(2•x/2) + 1

cos(x) + 1
Use the half angle formula cos x/2 =  ± sqrt( ( 1 + cos x ) / 2 ).
Use  cos(x) = 2 cos(x/2)\^2 – 1 ,

which follows from applying the Pythagorean identity to

cos(x) = cos(x/2)\^2 - sin(x/2)\^2  ,

which follows from the cosine addition formula.
I always just skip to Euler and use cos(x) = (e\^(ix) +e\^(-ix))/2.

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It's pretty straightforward to substitute that expression in for cosine and square it to derive the answer.
Just cos
Cos(2x) = 2cos²x  - 1,
For half angel that will be
cos(x) = 2cos²(x/2)  - 1 ,
Rearranging terms
2cos²(x/2) = cos(x) + 1