Trigonometric Identities

1Summary 重點摘要

Learn a few always-true trig formulas to simplify complicated trig expressions.

2Vocabulary cards Vocabulary

3Step by step Steps

1. What an identity isA trigonometric identity is an equation true for every angle θ — 30°, 45° or 120° alike; the two sides are always equal. This differs from an ordinary equation, which holds only for particular values. Because identities hold unconditionally, they may be substituted anywhere, to simplify complicated trig expressions or to prove two forms identical. At junior level only two identities are required; every other formula can later be derived from these two.
2. Identity one: sin²θ + cos²θ = 1The most important identity reads sin²θ + cos²θ = 1 for every angle. It grows out of Pythagoras' theorem: with sin θ = opposite ÷ hypotenuse and cos θ = adjacent ÷ hypotenuse, squaring and adding gives (opp² + adj²) ÷ hyp², and the numerator equals hyp², so the result is 1. For instance sin 30° = 0.5 and cos 30° ≈ 0.866, and 0.5² + 0.866² ≈ 0.25 + 0.75 = 1. Two rearrangements follow at once: sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ.
3. Identity two: tan θ = sin θ ÷ cos θThe second identity, tan θ = sin θ ÷ cos θ, extends the definition of tangent: tan θ = opposite ÷ adjacent, while sin θ ÷ cos θ = (opposite ÷ hypotenuse) ÷ (adjacent ÷ hypotenuse); the hypotenuses cancel in the division and the opposite ÷ adjacent remains, so the two agree. Thus tan θ may be rewritten as sin θ ÷ cos θ and back again at any time. For example tan 30° = 0.5 ÷ 0.866 ≈ 0.577, matching the table value of tan 30° and confirming the identity.
4. Simplifying with identitiesThe strategy is to spot familiar blocks and substitute them one at a time. Simplify sin²θ + cos²θ + tan θ × cos θ: the first block is exactly 1; in the second, replace tan θ with sin θ ÷ cos θ to get (sin θ ÷ cos θ) × cos θ, the cos θ cancels, leaving sin θ; the whole expression becomes 1 + sin θ. Always aim for the shortest possible form, and after finishing, scan the result for any block an identity can still reduce — simplify until nothing more can be done.
5. How to prove an identityA proof shows the two sides are identical. The standard route is to start from the more complicated side, substituting identities and simplifying algebraically until it matches the other side; do not work on both sides at once, nor assume the statement itself. To prove tan θ × cos θ = sin θ, handle only the left: tan θ × cos θ = (sin θ ÷ cos θ) × cos θ = sin θ, which equals the right side — proven. Change one thing per line and name the identity used at each key step.
6. Memory strategy and exam tipsOnly two formulas must be memorised: sin²θ + cos²θ = 1 and tan θ = sin θ ÷ cos θ. In a question, scan first for a squared-sum structure or a mix of tan with sin and cos — both signals point straight at the matching identity. Common errors include writing sin θ² (squaring θ before taking the sine) for sin²θ, and ignoring that cos θ cannot be zero inside a division. In proofs the marks lie in clear steps and skipped lines lose them; writing the two formulas at the top of your rough work before starting guards against forgetting mid-solution.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Two basics: sin²θ + cos²θ = 1
• tanθ = sinθ ÷ cosθ
• See sin² + cos² replace with 1
• See tanθ replace with sin ÷ cos
• Proof start from one side, simplify