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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.

    image0.png
  2. Rewrite each term.

    image1.png
  3. Replace each of the terms with an equivalent expression.

    image2.png

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

    image3.png
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