In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x=x" or usefully true, such as the Pythagorean Theorem's "a2+b2=c2" for right triangles. There are loads of trigonometric identities, but the following are the ones you're most likely to see and use.

Basic and Pythagorean Identities

\csc(x) = \dfrac{1}{\sin(x)}csc(x)=sin(x)1​

\sin(x) = \dfrac{1}{\csc(x)}sin(x)=csc(x)1​

\sec(x) = \dfrac{1}{\cos(x)}sec(x)=cos(x)1​

\cos(x) = \dfrac{1}{\sec(x)}cos(x)=sec(x)1​

\cot(x) = \dfrac{1}{\tan(x)} = \dfrac{\cos(x)}{\sin(x)}cot(x)=tan(x)1​=sin(x)cos(x)​

\tan(x) = \dfrac{1}{\cot(x)} = \dfrac{\sin(x)}{\cos(x)}tan(x)=cot(x)1​=cos(x)sin(x)​

Notice how a "co-(something)" trig ratio is always the reciprocal of some "non-co" ratio. You can use this fact to help you keep straight that cosecant goes with sine and secant goes with cosine.

The following (particularly the first of the three below) are called "Pythagorean" identities.

sin2(t) + cos2(t) = 1

tan2(t) + 1 = sec2(t)

1 + cot2(t) = csc2(t)

Note that the three identities above all involve squaring and the number1. You can see the Pythagorean-Thereom relationship clearly if you considerthe unit circle, where the angle ist, the "opposite" side issin(t) =y,the "adjacent" side iscos(t) =x, and the hypotenuse is1.