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!

someoneOctober 29, 2015 / 11:26 pmsinx^2-cosx^2=1

wrong… the identity is sinx^2+cos^2x=1

JacobMarch 20, 2016 / 12:30 pmI believe your first Pythagorean identity is wrong but that is coming from a freshman in college trig.

lydiaSeptember 20, 2016 / 6:31 pmBut sin^2x – cos^2x doesn’t equal one – – – sin^2x + cos^2x equals 1…

BobbiOctober 4, 2016 / 2:30 pmthe Pythagorean theorem states that a^2+b^2=c^2. Using the unit circle, we can substitute to get sin^2x+cos^2x=1. Where did you get the identity “sin^2x-cos^2x=1”? Using your identity #1, we would add cos^2x to each side to get the second identity, not subtract it, so if your identity #1 is correct, then your identity #2 would be “sin^2x=1+cos^2x”. Your steps are sound, but something is up with what you have listed as the correct identities.