What is the purpose of trigonometric identities

They exist. They simplify. Receive&understand them with joy&adulation.

 

What do you mean by "real life"? If you intend to spend the rest of your "real life" flipping burgers, then they are of no use at all. If you think you might, at some time in your life, be working in the sciences, they they are crucially important for simplifying a huge number of problems. If you are thinking of being an engineer, an auto mechanic, or a plumber, once again, problems involving trig functions or ratios are going to loom large, and those identities will simplify them. If you think you might be a lawyer, a business man, or a writer, science and engineering will still play a large part in your life. If you want to know what's going on there, it's a really good idea to know the basics of them- and trig functions and identities are part of the basics.

In any case, the purpose of education is not so much to teach you "facts" and "identities" as it is to teach you to think. The logical discipline involved in working with and proving trig identities is one form of that. (Just like insisting that you organize your thoughts, write "topic sentenses", cite facts, and give reference in research papers.)

 

proving is just the hard part and you really only need to do it once. It's not really proving them that is useful but actuallying using the trig idientities are what is useful.