The identity tan θ = sin θ / cos θ

Q: Do I need to memorize the tan θ = sin θ / cos θ identity for the exam?

No, this identity is explicitly provided on the official Edexcel 4PM1 formula sheet, so you do not need to memorize it, but you should know how to apply it correctly.

Q: When can I not use this identity?

The identity is only valid when , because division by zero is undefined. This means it cannot be used for values of where , such as or .

Edexcel International GCSE Further Pure Mathematics· 10F· 10 min read

1. What is the tan θ = sin θ / cos θ identity?★☆☆☆☆⏱ 3 min

📘 Definition

tan θ identity

A trigonometric identity stating that for all values of θ where $cos θ \neq = 0$ , $tan θ \equiv \frac{s i n θ}{c o s θ}$ . This identity is provided on the Edexcel 4PM1 formula sheet.

Example:

For $θ = 4 5^{\circ}$ , $sin 45 = \frac{2}{2}$ , $cos 45 = \frac{2}{2}$ , so $tan 45 = \frac{\frac{2}{2}}{\frac{2}{2}} = 1$ , which matches the known exact value.

The identity holds for all angles θ where cos θ is not zero, because division by zero is undefined. When cos θ = 0 (e.g., θ = 90°, 270°), tan θ is undefined, so the identity does not apply in these cases.

📐 Worked Example

Verify the identity holds for θ = 30° using exact trig values.

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Step 1: Write the exact values of sin 30°, cos 30° and tan 30°.
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$\sin 30^\circ = \frac{1}{2}, \cos 30^\circ = \frac{\sqrt{3}}{2}, \tan 30^\circ = \frac{1}{\sqrt{3}}$
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Step 2: Calculate the value of sin θ / cos θ for θ = 30°.
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$\frac{\sin 30^\circ}{\cos 30^\circ} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}$
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Step 3: Compare to tan 30°: the values are equal, so the identity holds for θ = 30°.

Exam tip:

Always check that cos θ ≠ 0 before using this identity, especially if you are working with angles close to 90° or 270°.

2. Simplifying trigonometric expressions using the identity★★☆☆☆⏱ 3 min

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One of the most common uses of this identity is simplifying expressions that mix tan, sin and cos terms. You can substitute tan θ with sin θ / cos θ, or rewrite a fraction of sin θ over cos θ as tan θ to reduce the number of terms in an expression.

📐 Worked Example

Simplify the expression $tan θ \times cos θ$ as much as possible.

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Step 1: Substitute tan θ with sin θ / cos θ using the identity.
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$\tan \theta \times \cos \theta = \frac{\sin \theta}{\cos \theta} \times \cos \theta$
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Step 2: Cancel the common cos θ terms in the numerator and denominator, provided cos θ ≠ 0.
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$\frac{\sin \theta}{\cancel{\cos \theta}} \times \cancel{\cos \theta} = \sin \theta$
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Step 3: The simplified expression is sin θ, valid for all θ where cos θ ≠ 0.

✓ Quick check

Test your understanding of simplification

What is the simplified form of $\frac{s i n θ}{t a n θ}$ ?
- cos θ
- sin θ
- tan θ
- 1

Exam tip:

If an expression has multiple tan terms, substituting all of them with sin/cos often allows you to cancel common factors and simplify the expression significantly.

3. Solving trigonometric equations using the identity★★★☆☆⏱ 4 min

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You can use the identity to solve equations that contain tan, sin and cos terms by rewriting all terms as sin and cos, or rearranging to group terms into the sin/cos ratio that can be replaced with tan.

📐 Worked Example

Solve the equation $2 sin θ = tan θ$ for 0° ≤ θ < 360°, where cos θ ≠ 0.

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Step 1: Substitute tan θ with sin θ / cos θ in the equation.
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$2 \sin \theta = \frac{\sin \theta}{\cos \theta}$
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Step 2: Rearrange all terms to one side and factor out the common sin θ term.
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$2 \sin \theta - \frac{\sin \theta}{\cos \theta} = 0 \implies \sin \theta \left(2 - \frac{1}{\cos \theta}\right) = 0$
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Step 3: Set each factor equal to zero and solve for θ.
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Case 1: sin θ = 0. Solutions in the range are θ = 0°, 180°.
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Case 2: 2 - 1/cos θ = 0 → 1/cos θ = 2 → cos θ = 1/2. Solutions in the range are θ = 60°, 300°.
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Step 4: Verify none of the solutions make cos θ = 0, all are valid. Final solutions: 0°, 60°, 180°, 300°.

Exam tip:

Never divide both sides of an equation by sin θ or cos θ unless you are certain they cannot be zero, as this will eliminate valid solutions. Always factor instead, as shown in the example.

4. Common Pitfalls

Wrong move:

Using the identity when cos θ = 0

Why:

Division by zero is undefined, so tan θ is undefined at these angles, making the identity invalid

Correct move:

Always check that cos θ ≠ 0 for your angle values before applying the identity

Wrong move:

Dividing both sides of an equation by sin θ or cos θ to simplify

Why:

This removes any solutions where the divided term equals zero, leading to incomplete answers

Correct move:

Rearrange all terms to one side and factor out the common trigonometric term instead of dividing

Wrong move:

Forgetting the identity is only valid for real angles where tan θ is defined

Why:

Applying it to angles like 90° will lead to mathematical errors

Correct move:

Exclude angles where cos θ = 0 (90°, 270° etc.) from your solution set when using this identity

Wrong move:

Rewriting cos θ / sin θ as tan θ

Why:

The identity states tan θ = sin θ / cos θ, not the reverse. Cotangent is out of scope for 4PM1 so you should not use that term either

Correct move:

Only replace a fraction of sin θ over cos θ with tan θ, leave cos θ / sin θ as is unless you can simplify it another way

Wrong move:

Memorizing the identity unnecessarily

Why:

It is provided on the formula sheet, so you waste memory space that could be used for non-provided formulae

Correct move:

Focus on learning to apply the identity correctly rather than memorizing it

5. Quick Reference Cheatsheet

Task	Action	Validity Note
Verify the identity	Substitute known sin/cos values and compare to tan	Only for θ where cos θ ≠ 0
Simplify expressions with tan	Replace tan θ with sin θ / cos θ, cancel common factors	Exclude θ where cos θ = 0
Solve equations with mixed tan/sin/cos	Substitute tan, rearrange and factor, solve for θ	Check no solutions have cos θ = 0

6. Frequently Asked

Do I need to memorize the tan θ = sin θ / cos θ identity for the exam?

No, this identity is explicitly provided on the official Edexcel 4PM1 formula sheet, so you do not need to memorize it, but you should know how to apply it correctly.

When can I not use this identity?

The identity is only valid when $cos θ \neq = 0$ , because division by zero is undefined. This means it cannot be used for values of $θ$ where $cos θ = 0$ , such as $θ = 9 0^{\circ}$ or $θ = 27 0^{\circ}$ .

Going deeper

formula_sheetEdexcel 4PM1 Official Formula SheetDownload the official formula sheet where this identity is provided

What's Next

Now that you have mastered applying the $tan θ = \frac{s i n θ}{c o s θ}$ identity, you are ready to build on this knowledge with other core trigonometric identities and problem-solving techniques for your Edexcel IGCSE Further Pure Math exam. Next, you will learn the Pythagorean identity $sin^{2} θ + cos^{2} θ \equiv 1$ , which is often used alongside the tan identity to simplify more complex expressions. Following that, you will cover trigonometric addition formulae and learn to solve a wider range of trigonometric equations, both of which regularly appear in 4PM1 exam papers. Make sure you practice applying this identity to mixed problems to reinforce your understanding before moving to more advanced topics.