Proving trigonometric identities
An identity is proved by transforming ONE side into the other — not by manipulating both sides of an assumed equality, which assumes what you are trying to show. The reliable strategies are: start from the more complicated side, convert everything to sine and cosine, combine fractions, and look for Pythagorean patterns to exploit.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Start with the messier sideThere is more to simplify, so you have more moves available.
- Convert to sine and cosineWhen the route is unclear, this almost always exposes it.
- Combine over a common denominatorThen simplify the numerator with sin² + cos² = 1 or a double-angle formula.
- Never operate on both sidesEach line must equal the previous one. Cross-multiplying an unproved equation is a logical error, not just bad style.
Worked example
Prove that (1 − cos 2θ)/sin 2θ = tan θ.
- Use cos 2θ = 1 − 2sin²θ, so 1 − cos 2θ = 2sin²θ.
- Use sin 2θ = 2 sin θ cos θ.
- The left side becomes 2sin²θ/(2 sin θ cos θ).
- Cancel 2 sin θ: = sin θ/cos θ.
Answer. = tan θ, so the identity holds (for sin θ ≠ 0 and cos θ ≠ 0).
Where marks get dropped
These are the specific errors that cost credit on proving trigonometric identities questions — QED's rubric penalises each of them separately.
- Working on both sides simultaneously. That assumes the identity, which is exactly what must be proved.
- Choosing the wrong form of cos 2θ. Picking 1 − 2sin²θ here made the numerator collapse instantly; cos²θ − sin²θ would have taken longer.
- Forgetting to note where the identity is undefined. Cancelling sin θ requires sin θ ≠ 0.
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Proving trigonometric identities — frequently asked questions
Can I work on both sides if I show each equals a third thing?
Yes — reducing both sides independently to a common expression is a valid and often clean structure. What is invalid is transforming an assumed equation.
What if I get stuck?
Convert everything to sine and cosine, combine fractions, and try multiplying by a conjugate such as (1 + cos θ). One of these almost always opens it up.
How is an identity different from an equation?
An identity holds for every value in the domain; an equation holds only for particular values. Proving an identity means proving it universally.
The rest of Trigonometry
Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Right-triangle ratios: sin, cos & tan
- 2The unit circle & exact values
- 3Radians, degrees & arc length
- 4Graphs of sin, cos & tan
- 5Pythagorean & reciprocal identities
- 6Angle-sum, difference & double-angle formulas
- 7Solving trigonometric equations
- 8The sine rule & the cosine rule
- 9Triangle area, sectors & segments
- 10Inverse trigonometric functions
- 11Proving trigonometric identities
- 12Polar coordinates & converting to Cartesian
- 13Modelling periodic phenomena with sinusoids
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