There are two trigonometric identities you must know for GCSE Higher: sin²θ + cos²θ = 1 (the Pythagorean identity) and tanθ = sinθ/cosθ. These are equations that are true for every angle θ, and they let you simplify expressions and prove other results without knowing the value of θ.
Where do the two identities come from?
Both identities follow directly from the definitions of sin, cos and tan in a right-angled triangle.
Consider a right-angled triangle with hypotenuse 1 (a "unit" triangle), angle θ at the base, opposite side = sinθ and adjacent side = cosθ.
Identity 1 — the Pythagorean identity:
By Pythagoras' theorem: (opposite)² + (adjacent)² = (hypotenuse)²
(sinθ)² + (cosθ)² = 1²
sin²θ + cos²θ = 1 ✓
Identity 2 — tan in terms of sin and cos:
By definition: tanθ = opposite/adjacent = sinθ/cosθ
tanθ = sinθ/cosθ ✓ (provided cosθ ≠ 0)
How do you use sin²θ + cos²θ = 1 to find a missing value?
The identity can be rearranged in two useful ways:
- sin²θ = 1 − cos²θ
- cos²θ = 1 − sin²θ
Worked example 1: Given that sinθ = 3/5 and θ is acute, find cosθ and tanθ.
Step 1: cos²θ = 1 − sin²θ = 1 − (3/5)² = 1 − 9/25 = 16/25
Step 2: cosθ = 4/5 (positive because θ is acute)
Step 3: tanθ = sinθ/cosθ = (3/5) ÷ (4/5) = 3/4
Check using a 3-4-5 right triangle: sinθ = 3/5, cosθ = 4/5, tanθ = 3/4. ✓
Worked example 2: Given that cosθ = −2/3 and 90° < θ < 180°, find sinθ.
Step 1: sin²θ = 1 − cos²θ = 1 − 4/9 = 5/9
Step 2: sinθ = √(5/9) = √5/3 (positive because sin is positive in the second quadrant)
How do you simplify a trigonometric expression using the identities?
Substitute the identities to replace one trig ratio with another.
Worked example 3: Simplify sin²θ + cos²θ + 3cos²θ.
sin²θ + cos²θ + 3cos²θ = 1 + 3cos²θ (using the Pythagorean identity)
Worked example 4: Simplify (sin²θ − 1)/cosθ.
Numerator: sin²θ − 1 = −(1 − sin²θ) = −cos²θ
So (sin²θ − 1)/cosθ = −cos²θ/cosθ = −cosθ
Worked example 5: Show that sin²θ/(1 − cos²θ) = 1.
Replace (1 − cos²θ) with sin²θ:
sin²θ/sin²θ = 1 ✓
How do you prove a trigonometric identity?
GCSE "prove" questions ask you to show that one expression equals another. Always work on one side only (usually the more complicated side) and reduce it to match the other.
Worked example 6: Prove that (1 − sin²θ)/cos θ ≡ cosθ.
Left-hand side: (1 − sin²θ)/cosθ
Replace (1 − sin²θ) with cos²θ:
= cos²θ/cosθ = cosθ = Right-hand side ✓
| Identity | Form you will use most |
|---|---|
| sin²θ + cos²θ = 1 | sin²θ = 1 − cos²θ OR cos²θ = 1 − sin²θ |
| tanθ = sinθ/cosθ | Replacing tanθ, or replacing sinθ/cosθ |
What are the most common exam mistakes?
- Writing sin²θ as (sinθ)² is correct notation — both mean the same thing. However, sinθ² (squaring only the θ) is incorrect.
- Confusing sin²θ + cos²θ = 1 with sinθ + cosθ = 1. The unsquared version is NOT an identity.
- Dividing by cosθ without checking it could be zero. In a proof, state "provided cosθ ≠ 0" or restrict θ appropriately.
- Working on both sides of the equation simultaneously in a "prove" question. Always pick one side and reduce it.
Frequently asked questions
Is sin²θ + cos²θ = 1 true for all angles, including obtuse angles?
Yes. Although the derivation above used a right-angled triangle with a hypotenuse of 1, the identity extends to all angles using the unit circle definition of sin and cos. For GCSE purposes, accept it as true for all values of θ and use it freely.
When would I use tanθ = sinθ/cosθ in a proof?
This identity is useful when an expression contains tanθ and you need to express everything in terms of sinθ and cosθ (or vice versa). It is also used to simplify tanθ × cosθ = sinθ — a result that appears in some exam questions.
Are there any other trig identities I need for GCSE?
No. Only sin²θ + cos²θ = 1 and tanθ = sinθ/cosθ are required at GCSE Higher. Further identities (sec, cosec, cot, double-angle formulae) are A-level content. At GCSE, every identity question can be answered using just these two.
How do I know whether to use the Pythagorean identity or the tan identity?
Look at what the question contains. If it has tanθ alongside sinθ or cosθ, use tanθ = sinθ/cosθ to convert. If it has a mix of sin² and cos² terms, use sin²θ + cos²θ = 1 (and its rearrangements) to eliminate one of them. Sometimes you need both identities in the same problem.
Work through GCSE trig identity problems with Professor Pi at aitutors.me.