An identity is a mathematical statement that is tautologically true. That is, it is true no matter what numerical value is assigned to it.
Here's an example of a simple one:
tanɵ = sinɵ/cosɵ.
Since we know from graphing some functions that the sine of an angle is equal to its y coordinate on the unit circle, and the cosine of an angle is equal to its x coordinate on the unit circle, we can then derive that our original formula tanɵ = opposite / adjacent can also be stated tanɵ = sinɵ/cosɵ, which is honestly a pretty elegant way to relating our three major trig functions to each other.
There are three other trig functions that I've been neglecting mentioning and I guess here is as good a place as any to introduce them. These are the inverse functions: cosecant, secant, and cotangent. They're just the multiplicative inverses of sine, cosine, and tangent - respectively. Thus:
cscɵ = 1/sinɵ
secɵ = 1/cosɵ
cotɵ = 1/tanɵ ... or ... cosɵ/sinɵ
See how that works? You just invert them. Also note that the prefix co- is present in the inverses of only the functions that don't already have one: sine & cosecant (abbreviated csc), cosine & secant (abbreviated sec), and of course tangent & cotangent (abbreviated cot). These are the most basic trig identities. These statements are true no matter what the value of ɵ is.
I'm not going to go through proofs for a bunch of identities here (not in this post anyway - I may do one or two later if I feel like it'll help my understanding) - there are resources that will help you with that if you want to pursue them. But I will give you another basic one, and one that pretty much blew my mind.
The name will probably be familiar to you, and for good reason. To the right you can see a lovely graph laying it out, but I'll take you through the proof. It's pretty elegant really, the way it plays out.
We start with the Pythagorean Theorem:
And finally, I'll leave you with the solution to last week's trivia.
tanɵ = sinɵ/cosɵ.
Now remember, the sine of any angle is equal to its y coordinate on the unit circle. The cosine is equal to its x coordinate on the unit circle. Now look at where the wave approaches infinity, and find those spots on the unit circle:
Illuminating, Isn't it?
To infinity and beyond! See you next time.