The following identities will be necessary for this problem:
•#sectheta = 1/costheta#
•#csctheta = 1/sintheta#
•#tantheta = sin theta/costheta#
•#cottheta = costheta/sintheta#
•#1 -sin^2theta = cos^2theta#
Now, we have what we need to prove:
#(1/costheta - sintheta/costheta)(1/sintheta + 1) = costheta/sintheta#
#((1 - sin theta)/costheta)((1 + sin theta)/sintheta) = costheta/sintheta#
#(1 - sin^2theta)/(costhetasintheta) = costheta/sintheta#
#cos^2theta/(costhetasintheta) = costheta/sintheta#
#costheta/sintheta = costheta/sintheta#
Identity proved!!
Practice exercises:
Prove the following identities:
a) #sintheta + cos^2theta/(1 + sin theta) = 1#
b) #sintheta(csctheta - sin theta) = cos^2theta#
exercises taken from http://www.swrschools.org/assets/algebra_2_and_trig/chapter12.pdf
Hopefully this helps and good luck!
