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The Pythagorean trigonometric identity is a trigonometric identity based on the application of the Pythagorean Theorem to the unit triangle.

The fundamental identity is:

Identity: , for all
Theorem. '
Prerequisites:
Pythagorean theorem:


Proof. Assume a point in the unit circle centered at the origin forming an angle . Let be a point on the x-axis corresponding to the x-coordinate of . Let be the origin. A right triangle would be formed: . The length of would be equal to and the length of would be equal to . The length of is 1 (since we're using the unit circle). Using the Pythagorean theorem, we have:
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Implications

Due to this fundamental relationship, other Pythagorean Identities emerge through the use of:

  • the complimentary and cofunction properties
  • the reciprocal functions
  • the quotient identities

The other identities include:

Identity:
Theorem. '
Proof.

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Identity:
Theorem. '
Proof.

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