Apply the reciprocal, quotient, and Pythagorean identities to simplify trigonometric expressions.
Three families of identities hold for every angle where both sides are defined. Reciprocal: csc=1/sin, sec=1/cos, cot=1/tan. Quotient: tan=sin/cos, cot=cos/sin. Pythagorean, built from the unit circle relationship sin^2(x)+cos^2(x)=1: dividing that equation by cos^2(x) gives 1+tan^2(x)=sec^2(x); dividing by sin^2(x) gives 1+cot^2(x)=csc^2(x).
Simplifying a trig expression means rewriting it, using these identities, as something shorter -- often a single function. The most reliable strategy is to rewrite everything in terms of sine and cosine FIRST, since every other function is built from those two; the simplification usually reveals itself once everything shares the same two building blocks.
Mastery looks like: They simplify a trig expression to a single function using the reciprocal, quotient, and Pythagorean identities, rewriting in sine and cosine as needed.
Common stumbles: Trying to simplify without first rewriting in terms of sine and cosine; misremembering which Pythagorean identity pairs with which reciprocal function.
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