Proving trig identities, one justified step at a time
Identity proofs reward strategy, not memory: convert to sine and cosine, work one side only, and look for a Pythagorean substitution.
What this covers
- Pythagorean identities
- Double- and half-angle formulas
- Sum and difference formulas
- Proof strategy and simplification
- Solving trig equations
Worked example
Problem
Prove that (1 − cos²θ)/sinθ = sinθ
- 1Recognise the identitysin²θ + cos²θ = 1, so 1 − cos²θ = sin²θ.
- 2SubstituteThe left side becomes sin²θ / sinθ.
- 3SimplifyCancel one factor of sinθ (valid for sinθ ≠ 0).
Answer: sinθ — identity proved
Mistakes that cost marks
- Working on both sides at once instead of transforming one side.
- Cancelling without noting the excluded values.
- Confusing the double-angle forms of cos 2θ.
Questions students ask
How many identities must I memorise?
Three Pythagorean plus the sum formulas. Everything else can be derived from them.
Practise trigonometry until it sticks
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