How would I verify this equality is accurate?

Question

(cosx + sinx) / (cosx - sinx) = (1+sin2x) / (cos2x)

thanks!

Answer 1

(cosx + sinx) / (cosx - sinx)
= (cos^x+2sinx cosx+sin^2x)/(cos^2x-sin^2x), times (cosx+sinx)/cosx+sinx)
= (1+sin2x) / (cos2x)

Answer 2

Multiply both numerator and denominator of LHS by (cosx+sinx).
This gives {cos^2x +2 cosxsinx +sin^2x}/(cos^2 -sin^2x)
Now make the following substitutions:
cos^2x +sin^2x =1
2cosxsinx =sin2x
cos^2x- sin^2x = cos2x
So LHS = (1+sin2x)/cos2x = RHS