How do you verify a trigonometric identity

An "identity" is a tautology, an equation or statement that is always true, no matter what. For instance, sin (x) = 1/csc (x) is an identity. To "prove" an identity, you have to use logical steps to show that one side of the equation can be transformed into the other side of the equation. You do not plug values into the identity to "prove" anything. There are infinitely-many values you can plug in. Are you really going to "prove" anything by listing three or four values where the two sides of the equation are equal? Of course not. And sometimes you'll be given an equation which is not an identity. If you plug a value in where the two sides happen to be equal, such as π/4 for the (false) identity sin (x) = cos (x), you could fool yourself into thinking that a mere equation is an identity. You'll have shot yourself in the foot. So let's don't do that.