Recipe

Verify a trig identity

Use when: you're given two trig expressions joined by an equals sign and asked to show they're the same for every \theta in the domain, not asked to solve for a specific \theta

Goal: one side rebuilt, one legal rewrite at a time, into something literally identical to the untouched other side

Taught in Simplifying and verifying trigonometric identities

1. 1

   Choose whichever side has more terms, more distinct function names, or a fraction, and commit to rewriting only that side. Treat the untouched side as the fixed destination you're rebuilding toward.

   Example

   For \tan\theta+\cot\theta=\sec\theta\csc\theta, the left side has two terms and two function names; the right side is already a single product. Work on the left.

   Why: an identity isn't yet known to be true, so any move that touches both sides at once (cross-multiplying, subtracting one side from the other) silently assumes the very fact still being proved.

2. 2

   Convert every function on your chosen side to sine and cosine using the reciprocal and quotient identities from earlier in this chapter.

   Example

   \tan\theta+\cot\theta = \dfrac{\sin\theta}{\cos\theta}+\dfrac{\cos\theta}{\sin\theta}.

   • two or more different function names are mixed together and nothing looks familiar yet — convert now, before anything else.
   • the side already shows a familiar pattern (like a Pythagorean substitution) without converting — skip ahead to Step 4 instead.

   Why: fewer distinct function names means fewer places to get stuck. Sine and cosine are the two building blocks every other trig function is defined from.

3. 3

   If more than one fraction is still present, factor each denominator, build the LCD, and rewrite every fraction over it before combining numerators. It's the same process from the rational-expressions chapter, just with denominators like \sin\theta or 1+\cos\theta instead of polynomials.

   Example

   \dfrac{\sin\theta}{\cos\theta}+\dfrac{\cos\theta}{\sin\theta} share no denominator yet; building each up to \sin\theta\cos\theta gives \dfrac{\sin^2\theta}{\sin\theta\cos\theta}+\dfrac{\cos^2\theta}{\sin\theta\cos\theta}=\dfrac{\sin^2\theta+\cos^2\theta}{\sin\theta\cos\theta}.

   • the fractions already share a denominator — combine the numerators directly.
   • the denominators differ — factor each one, build the LCD, rewrite each fraction over it, then combine.

   Why: a single fraction is something a Pythagorean substitution or a reciprocal identity can act on directly; several fractions with different denominators can't be compared or simplified further as they stand.

4. 4

   Scan the rewritten expression for a \sin^2\theta+\cos^2\theta pattern, or one of its rearrangements (1-\sin^2\theta, \sec^2\theta-1, and so on), ready to collapse to a single term.

   Example

   Step 3 above already produced \dfrac{\sin^2\theta+\cos^2\theta}{\sin\theta\cos\theta}, and the numerator is exactly \sin^2\theta+\cos^2\theta=1.

   • the pattern is already visible, in the numerator or after combining fractions — substitute it directly.
   • the pattern is hiding behind a denominator shaped like 1\pm\sin\theta or 1\pm\cos\theta, with no substitution visible yet — multiply that fraction by the conjugate over itself first (see the shortcut below), then look again.

   ⚡ Shortcut

   When the Pythagorean pattern isn't visible because it's hiding behind a denominator like 1-\sin\theta, multiply top and bottom by the conjugate, 1+\sin\theta: the product (1-\sin\theta)(1+\sin\theta)=1-\sin^2\theta manufactures the substitution instead of waiting to find it ready-made.

   Why: a Pythagorean substitution is the single move most likely to collapse two unrelated-looking terms into one, and it's easy to miss without deliberately searching for it.

5. 5

   Compare your rebuilt side to the untouched side term by term. "Close" or "equivalent-looking" isn't enough; every symbol has to match.

   Why: this is the entire point of the proof; the rewrite only counts as a verification once the two sides are the same expression, not merely related ones.