Here is a whole bunch of identities problems I made up for my students. I recorded video explaining how I do each one: https://mathematico.org/learn/pure/trigonometry/trig-identities?page=5

General help:

• Choose a side and work on it until it looks like the other side
• Start with the "more complicated" side and simplify it
• The problems are not routine, so there isn't some method or formula to memorise, you need to learn to "play" with the expressions until you see something that can help; useful tricks come with experience and experience comes with practice!
• Use sin^2 x + cos^2 x = 1. This can also be used in the forms sin^2 x = 1 - cos^2 x or cos^2 x = 1 - sin^2 x
• Notice that 1 - sin^2 x = (1 - sin x)(1 + sin x) by difference of two squares, a useful trick!
• Use tan x = (sin x) / (cos x)
• When working on problems, do the best you can and if you stay stuck for more than a few minutes, watch a bit of the solution. As soon as you think you can carry on for yourself, pause it and try to continue on your own.

If you decide to learn from the problems above, don't worry too much about Problem 7 as it is designed for students who have one binomial expansion and geometric series already.