I've been trying to prove the following identity:
(tan(x))/(sec(x)+1) = (sec(x)-1)/(tan(x))

For the left side, do you first have to multiply it by the conjugate?
I've tried numerous times and my paper just ends up being a huge mess, please help!

Convert the secant and tangent functions to their sine and cosine equivalents. Then you can simply by multiplying both sides by cos(x)/cos(x). Finally cross-multiply. What do you get?

sin(x)/(1+cos(x))
on the left side

and on the right side (1-cos(x))/(sin(x))

so now on the left side do I do the conjugate?

Cross multiply:

If a/b = c/d then ad = because