Hello, unfortunately trig identities take time and depend on the problem. Trig identities are like proofs where there is no one method that will solve all of them or even the next one. Only time and a lot of practice will help you understand it.
Here are some pointers I give my students that may help a little bit.
1. You almost always want to work with sin and cos when possible. If you see a csc, change it to 1/sin, if you see a sec change it to 1/cos, if you see tangent, change to sin / cos and if you see cot change it to cos / sin. Almost always immediately change these when you see them.
The exception to this s when you see any of these square or have a 1+ or 1 - next to them. Sometimes even then you should be less likely.
2. Memorize all the identities but definitely memorize the Pythagorean identities. Knowing these will help you substitute them when needed, Also note that although sin2x + cos2x = 1, this can become sin2x = 1 - cos2x, and cos2x = 1 sin2x. So you have to be alert for these as well. The Pythagorean identities are used in most identity problems. I'd say 65% of the time.
3. When you see fractions, look for common denominators and combine, because you would rather work with 1 faction than 2.
4. Make sure you know how to use conjugates of denominators to change it. Such as 2/ (1- sinx), if you multiply the top and bottom by 1 + sinx, you will get 2(1+sinx) / 1 - sin2x, which you can then change you denominator to cos2x.
5. This is kind of the opposite of 3. when you have 2 or more terms in the numerator, but one in the denominator, it might be okay to go in reverse and create 2 fractions where you can change them to nonfractinos after. example:
(sinx + cosx) / sinx can become sinx/sinx + cosx/sinx, which can become 1 + cotx.
Final note sometimes you will work on a problem for 5 minutes only to realize that you made a wrong turn and not any closer than before. THIS IS NORMAL. It happens to everyone and it would actually be weird if it didn't happen to you. Just start over and try again. It stinks but that is the nature of identities. We all had to suffer through them. At least it is Geometric proofs anymore!
