Question

1-2sin^2(theta/2)=2cos^2(theta/2)-1

Answer 1

To make it easier to type, let x = theta/2

1 - 2sin^2 (x) = 2cos^2 (x) - 1

1 - sin^2 (x) - sin^2 (x) = cos^2 (x) + cos^2 (x) - 1

Because sin^2 (x) + cos^2 (x) = 1:
1 - sin^2 (x) = cos^2 (x)
1 - cos^2 (x) = sin^2 (x)

cos^2 (x) - sin^2 (x) = cos^2 (x) - sin^2 (x)