How do you prove a trigonometric identity?

Treat identities as proofs, not equations

An identity claims that two expressions are equal for every value of the variable. Because you are trying to establish that the equality is true, you cannot treat it like an ordinary equation.

In an equation, you can add, subtract, multiply, or divide both sides by the same term. In an identity proof, doing that assumes the equality already holds before you have shown it. The core rule is to work on one side only until it transforms into the other.

A reliable three-step method

First, pick the more complicated side as your starting point. It gives you more terms, fractions, and operations to simplify.

Second, rewrite all functions in terms of sine and cosine. For example, replace tan(x) with sin(x)/cos(x). This turns trigonometry into algebraic fractions.

Third, look for fundamental identities such as sin^2(x) + cos^2(x) = 1 or double-angle formulas like sin(2x) = 2sin(x)cos(x). Remember that the Pythagorean identity works both ways: replacing 1 with sin^2(x) + cos^2(x) is often the step that reveals a hidden factorization.

A concrete example and pitfalls to avoid

Consider proving that tan(x)cos(x) = sin(x). Start with the left side: tan(x)cos(x). Replace tan(x) with sin(x)/cos(x). The expression becomes (sin(x)/cos(x)) * cos(x). The cos(x) factors cancel, leaving sin(x), which equals the right side.

Watch out for common algebra errors. Never cancel a term across an addition or subtraction sign; you may only cancel common factors. If your work stalls, stop and restart from the opposite side, as some proofs are much shorter from the other direction.