Cotangent and Cosecant Identities on a Unit Circle

Starting with the Pythagorean identity, sin²θ + cos²θ = 1, you can derive cotangent and cosecant Pythagorean identities. All you do is throw in a little algebra and apply the reciprocal and ratio identities and — poof! — two new identities.

1. Starting with the first Pythagorean identity, sin²θ + cos²θ = 1, divide each term by sin²θ.

   

2. Rewrite each term.

   

3. Replace each of the terms with an equivalent expression.

   

   Substituting these expressions into the equation and simplifying, you find that the result is