Question

2cos^2 22.5-1 (find exact value)

Answer 1

cos^2 θ/2 = (1 + cosθ)/2

cos^2 22.5° = (1 + cos 45°)/2
= (1 + √2/2)/2
= (2 + √2)/4

Answer 2

hmmph

It would help to notice that

2cos^2 θ - 1 = cos 2θ

so what you have is just cos 45° = 1/√2

note that 2(2+√2)/4 - 1 = 1+√2/2-1 = √2/2 = 1/√2