Worked Example: Trigonometric Identities

Calculus Readiness · Axiom Academy

EXAMPLE Verifying Trig Identities Prove an equation is an identity by transforming one side into the other — one legal move at a time. On the unit circle, a point P has coordinates . The horizontal leg, the vertical leg, and the radius form a right triangle whose hypotenuse is 1 . The Pythagorean theorem then reads . Almost every verification below is just this one fact in disguise — often used as . Verify the identity . Work on the left side and transform it into the right side. You will use four classic moves: multiply by a conjugate, apply the Pythagorean identity, cancel, and split a fraction. Nicely done. You verified by working only the left side — never moving a term across the equals sign. Conjugate trick: multiplying by turned the denominator into a difference of squares, . Pythagorean identity: collapsed the denominator and set up a clean cancellation. The four moves: nearly every identity yields to one or two of — common denominator, Pythagorean identity, double-angle identity ( ), or conjugate multiplication. Practice spotting those four moves and identity verification turns from a puzzle into a mechanical exercise — exactly the fluency calculus expects.

This is the written version of the interactive lesson above. See the full Calculus Readiness course.