Verifying Identities

Pre-Calculus · Axiom Academy

LESSON Verifying Trigonometric Identities Prove two trig expressions are equal by transforming one side — step by step — until it becomes the other. 1. What "Verify an Identity" Means Verifying is not solving. When you solve an equation you hunt for the specific x that makes it true. When you verify an identity you prove the two sides are always equal — so you leave one side fixed as the target and transform the other side until it lands exactly on it. …until it equals this fixed target 2. The Golden Rule: Work One Side Only It is tempting to treat the identity like an equation — cross-multiply, or add the same thing to both sides. Don't. Operating on both sides assumes they are already equal, which is exactly the thing you are trying to prove. A valid proof only ever rewrites one side using known identities. Transform one side step-by-step until it equals the other side. Add, subtract, multiply, divide, or cross-multiply across the = . Manipulating both sides equally is circular: it only preserves a true equation, and you have not yet shown this one is true. 3. First Move: Rewrite Everything in Sine and Cosine When you are stuck, convert every , , , and into sines and cosines. Once everything is built from the same two ingredients, fractions combine and factors cancel almost on their own.

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