Prove that cos x+sinx /cosx-sinx equal to 1+sin2x/cos2x

The expression you have provided is true because sin 2x =
2*sin x*cos x and cos 2x = (cos x)^2 - (sin x)^2

Let start
with (cos x + sin x)/(cos x - sin x)

multiply the numerator
and denominator by (cos x + sin x)

=> (cos x + sin
x)(cos x + sin x)/(cos x - sin x)(cos x + sin x)

=>
(cos x + sin x)^2/(cos x)^2 - (sin x)^2

=> [(cos
x)^2 + 2*sin x*cos x + (sin x)^2]/cos 2x

=> (1 + sin
2x)/cos 2x

This proves that (cos x + sin
x)/(cos x - sin x) = (1 + sin 2x)/cos 2x