Rewriting with an identity means replacing part of an expression with an equal trigonometric expression that makes the whole thing simpler. It appears in simplification questions, in “express in terms of” questions and at the start of most proofs and equations. The key is knowing the small set of identities and what each one lets you swap.
This lesson is part of trigonometric identities.
Which identities are you rewriting with?
Start from one fact: for any angle, sin²x + cos²x = 1. From it, three useful swaps follow.
- sin²x = 1 − cos²x
- cos²x = 1 − sin²x
- Divide the main identity by cos²x to get 1 + tan²x = sec²x, and by sin²x to get 1 + cot²x = cosec²x
You also need the definitions: tan x = sin x / cos x, sec x = 1 / cos x, cosec x = 1 / sin x and cot x = cos x / sin x.
How do you choose the swap?
- Decide the target. Does the question want one function only, or a simpler fraction?
- Find the square that can change. Squares of sin, cos, tan, sec, cosec and cot are the swap candidates.
- Replace it with the matching identity. Make the brackets explicit so the signs carry through.
- Expand and collect. Finish with ordinary algebra. If a quadratic in sin x appears, you can treat it like any quadratic (see the quadratic structure explorer).
Worked example
Express 5cos²θ + 2sin²θ in terms of sin θ only.
Step 1, target: only sin θ may remain, so cos²θ must be replaced.
Step 2, swap: cos²θ = 1 − sin²θ, so the expression is 5(1 − sin²θ) + 2sin²θ.
Step 3, expand: 5 − 5sin²θ + 2sin²θ.
Step 4, collect: 5 − 3sin²θ.
Check with θ = 30°: the original is 5 × 0.75 + 2 × 0.25 = 4.25, and the answer is 5 − 3 × 0.25 = 4.25. They agree.
The mistake to watch for
A common slip is to drop the square when using the identity.
Mistaken working: 1 − sin²x = cos x
The student remembered “1 minus sine squared is cosine” and lost the power.
Test it at x = 60°: 1 − sin²60° = 0.25, but cos 60° = 0.5. The correct statement is 1 − sin²x = cos²x, because 0.25 = 0.5². Whenever you rearrange sin²x + cos²x = 1, the squares stay on.
Check yourself
1. Write 4 − 4cos²x as a single term.
Show answer
4 − 4cos²x = 4(1 − cos²x) = 4sin²x.
2. Write 2tan²x + 3 in terms of sec x.
Show answer
tan²x = sec²x − 1, so 2(sec²x − 1) + 3 = 2sec²x − 2 + 3 = 2sec²x + 1.
3. Given sin θ = 3/5 and θ is acute, find cos θ and tan θ.
Show answer
cos²θ = 1 − 9/25 = 16/25. Since θ is acute, cos θ is positive, so cos θ = 4/5. Then tan θ = (3/5) ÷ (4/5) = 3/4.
Where this leads next
Once rewriting feels natural, move on to proving an identity without assuming the conclusion, which uses these same swaps line by line. The non-calculator working trainer is useful for practising the algebra without a calculator.
Some students follow each swap in class but freeze on a blank question, which is the gap our teachers work on in online one-to-one Additional Mathematics tuition.