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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