Question

How do you prove `cos(2x + pi) = cos^2 (x - pi/2) + cos(x + pi) sin(x + pi/2)`?

Answer 1

Use the trig unit circle as proof:

Left side ->`cos (2x + pi) = -cos 2x` (1)

Right side.

`cos^2 (x - pi/2) = sin^2 (x - pi/2)`
`cos (x + pi) = - cos x`
`sin(x + pi/2) = cos x`

Right side = `sin^2 x - cos^2 x = -cos 2x` = Left side.