The identity cos²θ + sin²θ = 1
Edexcel International GCSE Further Pure Mathematics· 10E· 15 min read
1. Derivation and Core Recall of the Identity★☆☆☆☆⏱ 3 min
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.
What is the value of sin²30° + cos²30°?
Reveal answer
1 —The identity holds for all angles, so the sum is always 1, no calculation needed!
2. Simplifying Trigonometric Expressions★★☆☆☆⏱ 4 min
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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.
Simplify the expression 2cos²θ + 3sin²θ
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Step 1: Rearrange the identity to substitute cos²θ = 1 - sin²θ
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Step 2: Expand the bracket
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Step 3: Combine like terms
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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.
3. Proving Trigonometric Results★★★☆☆⏱ 4 min
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.
Prove that (sinθ + cosθ)² - 1 ≡ 2 sinθ cosθ
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Step 1: Expand the squared bracket on the left-hand side (LHS)
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Step 2: Substitute the expanded form back into the LHS expression
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Step 3: Use the identity to replace sin²θ + cos²θ with 1
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Step 4: Simplify by cancelling the 1 and -1 terms
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4. Rewriting Equations to a Single Trigonometric Ratio★★★☆☆⏱ 4 min
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.
Rewrite the equation 3cos²θ = 2 + sinθ to use only sinθ as the trigonometric ratio
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Step 1: Rearrange the identity to replace cos²θ with a term in sin²θ: cos²θ = 1 - sin²θ
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Step 2: Expand the left-hand side
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Step 3: Rearrange into standard quadratic form, grouping all terms on one side
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This is now a quadratic equation in x = sinθ, which can be solved using the quadratic formula or factorisation if possible.
5. Common Pitfalls
Wrong move:
Writing cosθ² or sinθ² instead of cos²θ or sin²θ
Why:
This notation is ambiguous: it could mean cos(θ²) instead of (cosθ)², which will lose you marks in the exam.
Correct move:
Always write the squared sign after the trigonometric function abbreviation, e.g. cos²θ for (cosθ)².
Wrong move:
Forgetting the identity applies only to squared terms, trying to use it for cosθ + sinθ
Why:
cosθ + sinθ does NOT equal 1 for most values of θ, only the sum of their squares is always 1.
Correct move:
Only apply the identity when you see cos²θ or sin²θ terms, not linear sin or cos terms.
Wrong move:
Using out-of-scope identities like 1 + tan²θ = sec²θ for proofs or simplification
Why:
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 move:
Stick exclusively to cos²θ + sin²θ ≡ 1 and its direct rearrangements for all questions in this topic.
Wrong move:
Moving terms between sides of the equation during a proof question
Why:
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 move:
Work only on one side of the proof equation, using substitutions and algebraic expansion to show it equals the other side.
Wrong move:
Assuming the identity is given on the formula sheet, failing to memorize it
Why:
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 move:
Memorize the identity and its two rearrangements (cos²θ = 1 - sin²θ, sin²θ = 1 - cos²θ) before your exam.
6. Quick Reference 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 |
7. Frequently Asked
Is the cos²θ + sin²θ = 1 identity given on the Edexcel IGCSE FPM formula sheet?
No, you must memorize this identity: it is not included in the provided formula booklet for the 4PM1 specification.
Can I rearrange this identity to eliminate one trigonometric ratio?
Yes: use the rearrangements cos²θ ≡ 1 - sin²θ or sin²θ ≡ 1 - cos²θ to substitute out either squared ratio from any expression or equation.
Going deeper
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.