An identity is true for every angle, so you can swap one side of it for the other to simplify an expression. The most useful is sin²θ + cos²θ = 1.
This lesson is part of trigonometric functions. Later lessons such as solving equations over an interval rely on these rewrites.
Which identities should I know?
| Identity | Useful for |
|---|---|
| sin²θ + cos²θ = 1 | Swapping sin² for 1 − cos², or cos² for 1 − sin² |
| tan θ = sin θ ÷ cos θ | Turning tan into sin and cos |
| 1 + tan²θ = sec²θ | Simplifying expressions with sec or tan |
| 1 + cot²θ = cosec²θ | Simplifying expressions with cosec or cot |
The last two come from the first. Divide sin²θ + cos²θ = 1 by cos²θ to get tan²θ + 1 = sec²θ.
What should I try first?
A three-step habit works for most questions.
- Write everything in terms of sin and cos.
- Look for a factor or a common denominator.
- Replace sin² or cos² using sin²θ + cos²θ = 1, and cancel only common factors.
Worked example 1: a single fraction
Simplify (1 − cos²θ) ÷ (sin θ cos θ).
Replace 1 − cos²θ with sin²θ. The fraction becomes sin²θ ÷ (sin θ cos θ). Sin θ is a common factor of the whole top and the whole bottom, so cancel it: sin θ ÷ cos θ = tan θ.
Worked example 2: adding two fractions
Simplify sin θ ÷ (1 + cos θ) + (1 + cos θ) ÷ sin θ.
Use the common denominator sin θ(1 + cos θ):
[sin²θ + (1 + cos θ)²] ÷ [sin θ(1 + cos θ)]
Expand the bracket: sin²θ + 1 + 2cos θ + cos²θ. Since sin²θ + cos²θ = 1, the top becomes 2 + 2cos θ = 2(1 + cos θ).
The fraction is 2(1 + cos θ) ÷ [sin θ(1 + cos θ)]. The factor (1 + cos θ) is common to the whole top and bottom, so the result is 2 ÷ sin θ = 2 cosec θ.
Check with θ = 90°: the original is 1 ÷ 1 + 1 ÷ 1 = 2, and 2 cosec 90° = 2.
The mistake that costs marks
The common slip is to cancel a term that sits inside a sum. In (1 + cos θ) ÷ cos θ, a student cancels cos θ and writes 1, but cos θ is only part of the top.
| Step | Wrong | Right |
|---|---|---|
| Expression | (1 + cos θ) ÷ cos θ | (1 + cos θ) ÷ cos θ |
| Cancel | cos θ on top and bottom, giving 1 | Split: 1 ÷ cos θ + cos θ ÷ cos θ |
| Result | 1 | sec θ + 1 |
Test it: at θ = 60°, (1 + 0.5) ÷ 0.5 = 3, and sec 60° + 1 = 2 + 1 = 3, but the wrong answer 1 does not match. A quick number check catches this slip.
Check yourself
Simplify (cosec²θ − 1) tan²θ.
Answer
Use 1 + cot²θ = cosec²θ, so cosec²θ − 1 = cot²θ.
Then cot²θ × tan²θ = (1 ÷ tan²θ) × tan²θ = 1.
Check with θ = 45°: cosec²45° = 2, so (2 − 1) × 1 = 1.
What to study next
Identities are most useful inside equations. Go on to solving trigonometric equations over a stated interval, then test the chapter with the trigonometric functions practice set.
If you want a teacher to work through identities with you, see online one-to-one Additional Mathematics tuition.