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Answer 1 · 2017-04-04T21:21:42

$tanx+cosx/(1+sinx)=1/cosx$

Manipulate just the left-hand side.

$tanx+cosx/(1+sinx)=sinx/cosx+cosx/(1+sinx)$

Common denominator:

$=(sinx(1+sinx))/(cosx(1+sinx))+(cosx(cosx))/(cosx(1+sinx))$

$=(sinx+sin^2x+cos^2x)/((cosx)(1+sinx))$

Recall that $sin^2x+cos^2x=1$ :

$=(sinx+1)/(cosx(1+sinx))$

$=1/cosx$

Thus the identity is proven.