What are identities and how do you use them

I have a math final tomorrow, and need to know how to use identities and what they are. Could somebody please list them and say what they relate to?

Easy 10 points. Thanks!

Identities are equations with variables that hold true no matter what are substituted in for the variables. For example:

(x + y)^2 = x^2 + 2xy + y^2

is an identity, because it is satisfied for any x and y, which can be proven by expanding the LHS. An example of an equation that is not an identity:

2x + 1 = 1

This is not an identity because for some x, the LHS does not equal the RHS (e.g. when x = 1, LHS = 3, RHS = 1).

Of course, listing all identities is not feasible. There are infinitely many ways that we can make equations which are equal for any values of the variables.

The most common use of the word "identities" in high school is in the context trigonometric functions (the "trig identities"). These refer to (surprise, surprise) identities that involve trig functions. There are also infinitely many of these, however most of them can be derived from some fundamental identities (all angles in degrees):

sin^2(x) + cos^2(x) = 1
tan(x) = sin(x) / cos(x)
sin(90 - x) = cos(x)
sin(x + 360) = sin(x)
cos(x + 360) = cos(x)
sin(-x) = -sin(x)
cos(-x) = cos(x)
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x)tan(y))

How do we use them? Here's an example. If you know exact trig ratios, you may know that sin(30) = 1/2 (you can also check with your calculator). Using this fact, and the identity:

sin(90 - x) = cos(x)

we may calculate cos(60) without the use of a calculator. We just need to choose the appropriate value of x to substitute into the identity (which works for all x, since it is an identity). We want cos(60), so let's substitute x = 60:

cos(60) = sin(90 - 60) = sin(30)