# The identity cos²θ + sin²θ = 1

> Edexcel International GCSE Further Pure Mathematics · 4PM1 (2016 Higher Spec)
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s10-the-identity-cos-2-sin/

This guide covers the core Pythagorean trigonometric identity cos²θ + sin²θ ≡ 1, including recall, simplifying expressions, proving results, and rewriting equations to a single trig ratio aligned to 4PM1 requirements.

**Prerequisites:** [Basic right-angled triangle trigonometry (sin, cos definitions)](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s10-right-angled-trig/); [Exact trigonometric values for 30°, 45°, 60°](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s10-exact-trig-values/)

## Learning objectives

- Recall the identity cos²θ + sin²θ ≡ 1 (not provided on the exam formula sheet)
- Use the identity to simplify trigonometric expressions
- Apply the identity to prove results and rewrite equations using a single trigonometric ratio

## Derivation and Core Recall of the Identity

**Pythagorean Trigonometric Identity** — For any angle θ, the identity states cos²θ + sin²θ ≡ 1, and holds for all real values of θ.

*Example:* For θ = 45°, cos 45° = √2/2, sin 45° = √2/2, so (√2/2)² + (√2/2)² = 0.5 + 0.5 = 1, satisfying the identity.

This identity is derived directly from the Pythagorean theorem for a right-angled triangle with hypotenuse length 1 on the unit circle. The adjacent side to angle θ is cos θ, the opposite side is sin θ, so the sum of the squares of the sides equals the square of the hypotenuse: cos²θ + sin²θ = 1² = 1.

> **warning**
>
> This identity is NOT provided on your exam formula sheet. You must memorize it, along with its two rearrangements: cos²θ ≡ 1 - sin²θ and sin²θ ≡ 1 - cos²θ.

**Check your understanding**

1. What is the value of sin²30° + cos²30°?

   *Why:* The identity holds for all angles, so the sum is always 1, no calculation needed!

## Simplifying Trigonometric Expressions

You will frequently be asked to simplify expressions with mixed sin² and cos² terms using the identity. Substitute out one squared ratio to combine like terms.

**Worked example:** Simplify the expression 2cos²θ + 3sin²θ

1. Step 1: Rearrange the identity to substitute cos²θ = 1 - sin²θ
2. $$2(1 - \sin^2\theta) + 3\sin^2\theta$$
3. Step 2: Expand the bracket
4. $$2 - 2\sin^2\theta + 3\sin^2\theta$$
5. Step 3: Combine like terms
6. $$2 + \sin^2\theta$$
7. Note: You could also substitute sin²θ = 1 - cos²θ to get 3 - cos²θ, both are valid simplified forms.

> **Exam tip:** Choose the substitution that reduces the number of terms in your expression, or matches the form requested in the question.

*Calculator:* allowed

## Proving Trigonometric Results

Proof questions require you to show that one side of an equation is equal to the other using only valid algebraic manipulations and the identity. Never move terms between sides of the equation during a proof: work only on one side until it matches the other.

**Worked example:** Prove that (sinθ + cosθ)² - 1 ≡ 2 sinθ cosθ

1. Step 1: Expand the squared bracket on the left-hand side (LHS)
2. $$(\sin\theta + \cos\theta)^2 = \sin^2\theta + 2\sin\theta\cos\theta + \cos^2\theta$$
3. Step 2: Substitute the expanded form back into the LHS expression
4. $$LHS = (\sin^2\theta + \cos^2\theta) + 2\sin\theta\cos\theta - 1$$
5. Step 3: Use the identity to replace sin²θ + cos²θ with 1
6. $$LHS = 1 + 2\sin\theta\cos\theta - 1$$
7. Step 4: Simplify by cancelling the 1 and -1 terms
8. $$LHS = 2\sin\theta\cos\theta = RHS, as required$$

> **tip**
>
> If you get stuck on a proof, look for squared sin or cos terms: these are your cue to use the cos²θ + sin²θ ≡ 1 identity.

## Rewriting Equations to a Single Trigonometric Ratio

To solve trigonometric equations with both sinθ and cosθ terms, you can use the identity to rewrite the equation so it only contains one ratio, which you can then solve using standard techniques.

**Worked example:** Rewrite the equation 3cos²θ = 2 + sinθ to use only sinθ as the trigonometric ratio

1. Step 1: Rearrange the identity to replace cos²θ with a term in sin²θ: cos²θ = 1 - sin²θ
2. $$3(1 - \sin^2\theta) = 2 + \sin\theta$$
3. Step 2: Expand the left-hand side
4. $$3 - 3\sin^2\theta = 2 + \sin\theta$$
5. Step 3: Rearrange into standard quadratic form, grouping all terms on one side
6. $$3\sin^2\theta + \sin\theta - 1 = 0$$
7. This is now a quadratic equation in x = sinθ, which can be solved using the quadratic formula or factorisation if possible.

## Common pitfalls

- **Wrong:** Writing cosθ² or sinθ² instead of cos²θ or sin²θ
  - Why it fails: This notation is ambiguous: it could mean cos(θ²) instead of (cosθ)², which will lose you marks in the exam.
  - Correct: Always write the squared sign after the trigonometric function abbreviation, e.g. cos²θ for (cosθ)².
- **Wrong:** Forgetting the identity applies only to squared terms, trying to use it for cosθ + sinθ
  - Why it fails: cosθ + sinθ does NOT equal 1 for most values of θ, only the sum of their squares is always 1.
  - Correct: Only apply the identity when you see cos²θ or sin²θ terms, not linear sin or cos terms.
- **Wrong:** Using out-of-scope identities like 1 + tan²θ = sec²θ for proofs or simplification
  - Why it fails: These identities are not part of the 4PM1 syllabus, and examiners expect you to use only the cos²θ + sin²θ ≡ 1 identity for these questions.
  - Correct: Stick exclusively to cos²θ + sin²θ ≡ 1 and its direct rearrangements for all questions in this topic.
- **Wrong:** Moving terms between sides of the equation during a proof question
  - Why it fails: Proof questions require you to manipulate one side only to match the other; moving terms between sides assumes the identity is true before you prove it, which is logically invalid.
  - Correct: Work only on one side of the proof equation, using substitutions and algebraic expansion to show it equals the other side.
- **Wrong:** Assuming the identity is given on the formula sheet, failing to memorize it
  - Why it fails: The cos²θ + sin²θ ≡ 1 identity is not included in the 4PM1 formula booklet, so you will not be able to look it up during the exam.
  - Correct: Memorize the identity and its two rearrangements (cos²θ = 1 - sin²θ, sin²θ = 1 - cos²θ) before your exam.

## Cheatsheet

| Identity | Key Rearrangements | Use Cases |
| --- | --- | --- |
| cos²θ + sin²θ ≡ 1 | cos²θ ≡ 1 - sin²θ, sin²θ ≡ 1 - cos²θ | Simplify expressions, prove results, rewrite equations to single trig ratio |
| Critical Note | NOT given on formula sheet (must recall) | Only applies to squared trigonometric terms, not linear terms |

## What's next

Now that you have mastered the cos²θ + sin²θ ≡ 1 identity, you are ready to move on to related trigonometry topics in the Edexcel IGCSE FPM syllabus. This identity is a foundational tool that you will use frequently across trigonometry questions, including when solving equations and working with double-angle formulae later in the unit. Make sure you practice applying the identity to a range of expression simplification and proof questions to build confidence before your exam, as it is tested in almost every year's paper.

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