Tangent and Secant Identities on a Unit Circle

By Mary Jane Sterling

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