Trigonometric identities

A trigonometric identity is an equation that is true for every value of the angle where both sides make sense. In Additional Mathematics you use identities to simplify expressions, prove statements and rewrite a question into something you can solve. Check the current 0606 syllabus on the Cambridge page for exactly which identities your examination year lists.

What do you need before you start?

You should be comfortable with the sine, cosine and tangent ratios in a right-angled triangle, and with their values for angles beyond 90°. Radians, arcs and sectors is the module just before this one, and the same identities work whether the angle is in degrees or radians.

You also need algebra fluency: common denominators, expanding brackets and factorising. Most identity proofs are algebra wearing trigonometric labels.

What does an identity question look like?

Here is one orienting example. Simplify (1 − cos²x) / (sin x cos x).

1. The numerator matches a rearranged identity: sin²x + cos²x = 1, so 1 − cos²x = sin²x.
2. The fraction becomes sin²x / (sin x cos x).
3. Cancel one common factor of sin x from the numerator and denominator: sin x / cos x.
4. Use tan x = sin x / cos x, so the answer is tan x.

Step 3 is only allowed when sin x ≠ 0, and the original fraction needs cos x ≠ 0 as well. A quick check at x = 30°: the original gives 0.25 / (0.5 × 0.866) ≈ 0.577, and tan 30° ≈ 0.577.

In what order should you study the lessons?

1. Use a fundamental identity to rewrite an expression: start here, because every later skill depends on recognising sin²x + cos²x = 1 and its two relatives.
2. Prove an identity without assuming its conclusion: the method for “show that” questions, working on one side only.
3. Simplify an expression before substitution: how an identity turns a hard evaluation into a short one.
4. State excluded values in a trig identity: where an identity stops being valid, and why it matters.
5. Diagnose an invalid cancellation: the algebra error that costs the most marks in this topic.

Then try the mixed practice set. Keep a record of which step went wrong each time with the mistake log and retest queue.

What are the common traps?

• Dropping the square. Writing 1 − sin²x = cos x instead of cos²x.
• Cancelling terms instead of factors. (sin x + cos x) / cos x is not sin x + 1.
• Working from both sides at once. A proof must show the left side becomes the right side, line by line.
• Forgetting excluded values. Dividing by cos x quietly assumes cos x ≠ 0.
• Treating tan x = 3 as sin x = 3. The ratio is 3, not the sine.

How should you use the practice set?

Do the questions in order on paper, without notes, and write a short test value (such as x = 30°) beside any simplification you are unsure about. Open the answer only after you have an answer of your own. The routing table at the end of the set tells you which lesson to revisit for each kind of slip.

If this module sits inside a bigger revision plan, the Additional Mathematics learning guide shows where it fits, and the next module, trigonometric equations and graphs, uses these identities to solve equations. Students who can follow the examples but stall on a blank question can read how we teach that gap in online one-to-one Additional Mathematics tuition.