Question

How do you prove `(cosx-sinx)/(cosx+sinx)=(cos2x)/(1+sin2x)`?

Answer 1

Prove trig equation

Explanation:

`(cos x - sin x)/(cos x + sin x) =`
`= [(cos x - sin x)/(cos x + sin x)] [cos x + sin x)/(cos x + sin x)] =`
`=(cos^2 x - sin^2 x)/(cos x + sin x)^2 =`
Apply trig identities:
`cos^2 x - sin^2 x = cos 2x`
`(cos x + sin x)^2 = cos^2 x + 2sin x.cos x + sin^2 x = 1 + sin 2x`
Finally,
`(cos x - sin x)/(cos x + sin x) = (cos 2x)/(1 + sin 2x)`