Let's work with the left hand side 
We're going to times cosx/1-sinx by 1+sinx/1+sinx, you can do this because 1+sinx/1+sinx is equal to 1, so you aren't really changing anything. 

After you've times the top and bottom by 1+sinx, you get: 

cosx(1+sinx) / (1-sinx)(1+sinx) 

The denominator is a difference of squares, so you get 1-sin^2 x 

what you have now is 

cosx(1+sinx) / 1-sin^2x 

You know that 1-sin^2x is equal to cos^2x 

so know you have 

cosx(1+sinx) / cos^2x 

get rid of the cosx on top by simplifying and the cos^2x so that you're left with 

(1+sinx) / cos x 

Therefore cosx / 1- sinx = 1 + sinx / cos x