I thought I’d post this quick simplification using trig identities since it gave me a bit of a run for my money. Partially so I remember it better and second, to lend a bit of help to those who might be stuck.

**The Problem:** Simplify

Now, how hard could this be?!? As it turns out, it’s not too bad, but the trick is in factoring what we have.

**Factor things out:**

Great! Now that we have this in better form we need to bring in a couple of trig identities which will help us solve this and one that I’m going to cite because it could trick you!

**Pythagorean Identity 1:**

**Pythagorean Identity 2:**

**Pythagorean Identity 3:** (*This is the tricky one! As much as it seems like we could use this identity we would end up loosing terms. Don’t use this*)

Now, things should start looking a bit simpler to solve. So, here we go starting with what we factored:

1)

2) Use Identity 1 to substitute for the first term:

3) Use Identity 2 to substitute for the first part of the second term:

4) Now combine the two like terms:

5) And so we have it! The final result is:

Now, looking back, if you notice in identity 3, if you use it, you end up with a term which multiplied out you loose the negative and the 2 in front of cos. This is a big problem because if you did it this way, what you get in the end *does not equal* what you had in the beginning and thus, isn’t a valid way to simplify this problem. As a result, you must use identities 1 and 2 as shown in the 5 steps above.

Cheers!