Double-Angle Identities

Calculus Readiness · Axiom Academy

LESSON Double-Angle Identities One rule, born by setting the two angles equal: how , , and unfold from the sum formulas. Start with the sum identities you already know. They take two angles, A and B . The double-angle identities are just what happens when those two angles are the same — slide A until it lands on , and A+B becomes . The sum identities — two independent angles Put into and the two cross-terms are identical, so they add: The animation makes the equality concrete. On the left, the point at has height . On the right, that same height is built as the product — two copies of a piece. Same number, two stories. You will read this identity both directions: collapse into to integrate it, or expand into to differentiate or factor. Recognizing the form is the whole skill. 3. Cosine — Three Forms, One Identity Cosine has three equivalent double-angle forms. They are not three facts — they are one fact, rewritten twice using the Pythagorean identity . Watch the bar trade places with , then the bar trade places with . Pure cosine — feeds straight into the power-reduction formula for . Pure sine — feeds straight into the power-reduction formula for . 4. Tangent, and Using the Identities Tangent comes from the same move on : This form is undefined exactly when , i.e. — and at those angles , where is genuinely undefined. The formula's failure is the truth, not a flaw.

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