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.

    image0.png
  2. Rewrite each term.

    image1.png

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

    image2.png
  3. Replace each of the terms with an equivalent expression.

    image3.png

    Substituting these expressions into the equation and simplifying, you get

    image4.png
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