How to prove a trigonometric identity

Proving a trigonometric identity means showing that two expressions are equal for all valid angles, not just solving for one. In the Form 5 Trigonometric Functions chapter of Add Math, these questions ask you to 'show that' or 'prove that' the left-hand side equals the right-hand side.

The method is to start on one side and rewrite it, step by careful step, using algebra and the standard identities, until it matches the other side exactly. Unlike solving an equation, you are not finding a value of the angle, you are demonstrating that the statement is always true.

Each clean line of working earns method marks, so a tidy, logical chain matters as much as the final line.

Use this method whenever a question says 'prove that' or 'show that' an expression involving $\sin$, $\cos$, $\tan$, $\sec$, $\operatorname{cosec}$ or $\cot$ equals another expression. The tell-tale sign is an equals sign with a full expression on each side and no range given, you are not solving for an angle, so there is nothing to 'find'.

Look at both sides first: whichever is more complicated, or contains fractions, is usually the side to start from. If you see $1-\sin^{2}\theta$, $1+\tan^{2}\theta$ or a sum of fractions, a Pythagorean identity or a common denominator is almost certainly the next move.

1. 1

   Pick one side

   Choose the more complicated side to work on, usually the one with fractions or higher powers.

2. 2

   Combine fractions

   If there are fractions, add them over a common denominator.

3. 3

   Simplify the numerator

   Expand and collect like terms in the numerator carefully.

4. 4

   Apply an identity

   Use a Pythagorean identity such as $1-\sin^{2}\theta=\cos^{2}\theta$ to simplify.

5. 5

   Convert to target ratios

   Rewrite in the functions that appear on the other side, e.g. $\frac{1}{\cos^{2}\theta}=\sec^{2}\theta$.

6. 6

   Conclude

   State that this side now equals the other side, so the identity is proved.

• Treating the identity like an equation and moving terms across the equals sign, this assumes what you are trying to prove.
• Expanding $(1+\sin\theta)(1-\sin\theta)$ incorrectly instead of using the difference of squares $1-\sin^{2}\theta$.
• Using the wrong Pythagorean identity, e.g. writing $1-\sin^{2}\theta=\cos\theta$ instead of $\cos^{2}\theta$.
• Confusing $\sec\theta$ with $\frac{1}{\sin\theta}$, remember that $\sec\theta=\frac{1}{\cos\theta}$.
• Stopping one step early, before the side you are working on matches the other side exactly.

The habit that costs the most marks is crossing the equals sign, students multiply both sides or shuffle terms, which quietly assumes the result. In a one-to-one lesson our teachers keep you on a single side and ask 'what does this side become next?', so the logic stays honest and every line follows from the last.

We also drill the three Pythagorean identities until choosing the right one is instant. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

To see how we teach identity proofs, message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.