Pythagorean Identities

Pre-Calculus · Axiom Academy

The most fundamental trig identities — all three fall right out of the Pythagorean theorem on the unit circle. 1. The Identity, Read Off the Circle Put a point at angle on the unit circle. Its coordinates are , so the horizontal leg has length , the vertical leg has length , and the hypotenuse is the radius — exactly 1 . The Pythagorean theorem on this triangle says the two leg-squares add up to the hypotenuse-square. The point's coordinates on the unit circle Pythagoras on the legs , and hypotenuse 1 2. Divide by : the Tangent–Secant Form An identity stays true if we divide both sides by the same nonzero quantity. Divide through by . Geometrically this rescales the whole triangle until its base is 1 : the vertical leg becomes and the hypotenuse becomes . — the vertical leg scales to . — the base scales down to exactly 1 . — Pythagoras on the rescaled triangle. Dividing every side length by is a similarity transform — angles are preserved, so it's still a right triangle. The base becomes , which is why the identity reads . 3. Divide by : the Cotangent–Cosecant Form Take the very same identity and divide through by instead. Now the triangle rescales until its vertical leg is 1 : the base becomes and the hypotenuse becomes . The term: — the base scales to . The right-hand side: — the hypotenuse scales to , giving . All three Pythagorean identities come from one right triangle on the unit circle — then dividing by or . Scroll up to revisit any step.

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