1. Verifying a trig identity means using the various ratio and Pythagorean identities in order to simplify long and complicated problems with carious trig functions into something as simple as sin(x). It also means verifying that two different trig ratios are equal to each other which again means simplifying. So all in all verifying trig identities mainly means to simplify big long complicated problems into simple trig functions that are easy to work with.
2. There aren't many tips and tricks to solving these identities. One of the main tips I have is to get everything into related trig functions in order to be able to use a identity. Also you should probably never square both sides because itll make the problem more complicated than it already is.
3. The first thing I do is see if the trig functions currently relate to each other. If they do I look at the multiple possible path and choose the easiest one. After that i keep going using multiple trig identities or factoring along the way if i can. When its to verify I do it until i get both sides equal to each other. When its to simplify I go until there is hopefully just one trig function left