How to Prove Complex Identities by Working Individual Sides of a Trig Proof
Sometimes doing work on both sides of a trig proof, one side at a time, leads to a quicker solution. This is because in order to prove a very complicated identity, you may need to complicate the expression even further before it can begin to simplify. However, you should take this action only in dire circumstances after every other technique has failed.
The main idea is that you work on the left side first, stop when you just can’t go any further, and then switch to working on the right side. By switching back and forth, your goal is to make the two sides of the proof meet in the middle somewhere.
For example, follow the steps to prove this identity:

Break up the fraction by writing each term in the numerator over the term in the denominator, separately.
The rules of fractions state that when only one term sits in the denominator, you can do this step because each part on top is being divided by the bottom.
You now have

Use reciprocal rules to simplify.
The first fraction on the left side is the reciprocal of
You now have
You’ve come to the end of the road on the left side. The expression is now so simplified that it would be hard to expand it again to look like the right side, so you should turn to the right side and simplify it.

Look for any applicable trig identities on the right side.
You use a Pythagorean identity to identify that
You now have

Cancel where possible.
Aha! The right side has
which is 0! Cancel them to leave only

Rewrite the proof starting on one side and ending up like the other side.
Keep in mind that some precalculus teachers do not accept working on both sides of an equation as a valid proof. If you’re unfortunate enough to encounter a teacher like this, you should still work on both sides of the equation, but for your eyes only. Be sure to rewrite your work for your teacher by simply going down one side and up the other (like you did in Step 5).