Pearson Edexcel · A-Level · Mathematics
Course version: 9MA0 · Version from Mar 2026
Understand and use tan θ = sin θ / cos θ. Understand and use sin²θ + cos²θ…
Full specification wording
Trigonometry: 5.5 Understand and use tan θ = sin θ / cos θ. Understand and use sin²θ + cos²θ = 1, sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ.
The identity tan θ = sin θ / cos θ defines tangent in terms of sine and cosine and is valid wherever cos θ ≠ 0.
Checked against the Pearson Edexcel 9MA0 specification (opens in a new tab) on .
- Point 5 of 9
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
The identity tan θ = sin θ / cos θ defines tangent in terms of sine and cosine and is valid wherever cos θ ≠ 0. The Pythagorean identity sin²θ + cos²θ = 1 holds for all θ and is the foundation for the other two identities. Dividing sin²θ + cos²θ = 1 by cos²θ gives tan²θ + 1 = sec²θ, usually written sec²θ = 1 + tan²θ, valid where cos θ ≠ 0. Dividing by sin²θ gives 1 + cot²θ = cosec²θ, usually written cosec²θ = 1 + cot²θ, valid where sin θ ≠ 0. These identities are used to simplify expressions, prove other identities, and solve trigonometric equations.
For example: To solve 2sin²θ + 3cos²θ = 2, replace sin²θ with 1 − cos²θ to obtain 2 + cos²θ = 2, so cos²θ = 0 and θ = π/2 + kπ.
What earns the marks
- State and use tan θ = sin θ / cos θ, noting it is undefined where cos θ = 0.
- State and use sin²θ + cos²θ = 1 for all θ.
- Derive sec²θ = 1 + tan²θ by dividing sin²θ + cos²θ = 1 by cos²θ.
- Derive cosec²θ = 1 + cot²θ by dividing sin²θ + cos²θ = 1 by sin²θ.
- Use the identities to simplify expressions such as (1 − cos²θ)/sin θ or to solve equations.
- State restrictions on θ where denominators are zero, for example when using sec²θ = 1 + tan²θ.
Common mistakes and how to put them right
Writing sin²θ + cos²θ = 1 as sin θ + cos θ = 1.
Put it right: The squares apply to the function values, not the angle.
Forgetting that sec²θ = 1 + tan²θ is only valid where cos θ ≠ 0, and cosec²θ = 1 + cot²θ only where sin θ ≠ 0.
Incorrectly rearranging sec²θ = 1 + tan²θ as sec θ = 1 + tan θ.
Put it right: The identity relates the squares, not the functions themselves.
Exam tips
- When proving an identity, start from the more complicated side and simplify using sin²θ + cos²θ = 1.
- To solve equations involving sec²θ or cosec²θ, convert to tan²θ or cot²θ using the Pythagorean identities.
- Always check whether a given value of θ makes a denominator zero before using an identity that involves division.
Try the lesson
The start of the Trigonometry: 5.5 Understand and use tan θ = sin θ / cos θ. Understand and use sin²θ + cos²θ = 1, sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ. lesson, made from checked teaching notes for this specification point.
Have a go first
Given that sin θ = 3/5 and cos θ = 4/5, find the exact value of tan θ.
Listen
The identity tan θ = sin θ / cos θ defines tangent in terms of sine and cosine and is valid wherever cos θ ≠ 0. The Pythagorean identity sin²θ + cos²θ = 1 holds for all θ and is the foundation for the other two identities. Dividing sin²θ + cos²θ = 1 by cos²θ gives tan²θ + 1 = sec²θ, usually written sec²θ = 1 + tan²θ, valid where cos θ ≠ 0. Dividing by sin²θ gives 1 + cot²θ = cosec²θ, usually written cosec²θ = 1 + cot²θ, valid where sin θ ≠ 0. These identities are used to simplify expressions, prove other identities, and solve trigonometric equations. For example, to solve 2sin²θ + 3cos²θ = 2, replace sin²θ with 1 − cos²θ to obtain 2 + cos²θ = 2, so cos²θ = 0 and θ = π/2 + kπ. A strong answer earns marks for these points. State and use tan θ = sin θ / cos θ, noting it is undefined where cos θ = 0. State and use sin²θ + cos²θ = 1 for all θ. Derive sec²θ = 1 + tan²θ by dividing sin²θ + cos²θ = 1 by cos²θ. Derive cosec²θ = 1 + cot²θ by dividing sin²θ + cos²θ = 1 by sin²θ. Exam tip: When proving an identity, start from the more complicated side and simplify using sin²θ + cos²θ = 1.
Key words
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Fundamental trigonometric identities
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Derivation of Pythagorean identities
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Simplification and proof in trigonometry
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Solving trigonometric equations using identities
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See every definition
- Fundamental trigonometric identities
- Identities including tan θ = sin θ / cos θ and sin²θ + cos²θ = 1, used to simplify expressions, prove other identities and solve trigonometric equations.
- Derivation of Pythagorean identities
- Dividing sin²θ + cos²θ = 1 by cos²θ gives sec²θ = 1 + tan²θ, and dividing by sin²θ gives cosec²θ = 1 + cot²θ.
- Simplification and proof in trigonometry
- Using the identities to simplify expressions and prove other identities, starting from the more complicated side and stating any restrictions on θ.
- Solving trigonometric equations using identities
- Replacing terms such as sin²θ with 1 − cos²θ so an equation can be solved, checking whether a value of θ makes a denominator zero.
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