Tangent and Secant Identities on a Unit Circle

Starting with the Pythagorean identity, sin²θ + cos²θ = 1, you can derive tangent and secant 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 cos²θ.

   

2. Rewrite each term.

   

   Note how the exponent 2 is pulled out of the parentheses:

   

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

   

   Substituting these expressions into the equation and simplifying, you get