# The identity tan θ = sin θ / cos θ

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

This guide teaches you to apply the $\tan \theta = \frac{\sin \theta}{\cos \theta}$ identity for Edexcel IGCSE Further Pure Math (4PM1). You will learn to simplify trig expressions and solve basic equations using this formula, which is provided on your exam formula sheet.

**Prerequisites:** [Basic sine, cosine and tangent definitions for right-angled triangles](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s10-trigonometric-ratios/); [Exact trig values for 30°, 45°, 60°](https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s10-exact-trig-values/)

## Learning objectives

- Recall the identity tan θ = sin θ / cos θ and note it is provided on the 4PM1 formula sheet
- Simplify trigonometric expressions using the tan identity
- Solve basic trigonometric equations by rewriting tan terms using the identity

## What is the tan θ = sin θ / cos θ identity?

**tan θ identity** — A trigonometric identity stating that for all values of θ where $\cos \theta \neq 0$, $\tan \theta \equiv \frac{\sin \theta}{\cos \theta}$. This identity is provided on the Edexcel 4PM1 formula sheet.

*Example:* For $\theta = 45^\circ$, $\sin 45 = \frac{\sqrt{2}}{2}$, $\cos 45 = \frac{\sqrt{2}}{2}$, so $\tan 45 = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$, 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.

> **info**
>
> You do not need to memorize this identity for your exam, as it is printed on the official formula sheet. However, you should recognize it immediately and know when to apply it.

**Worked example:** Verify the identity holds for θ = 30° using exact trig values.

1. Step 1: Write the exact values of sin 30°, cos 30° and tan 30°.
2. $$\sin 30^\circ = \frac{1}{2}, \cos 30^\circ = \frac{\sqrt{3}}{2}, \tan 30^\circ = \frac{1}{\sqrt{3}}$$
3. Step 2: Calculate the value of sin θ / cos θ for θ = 30°.
4. $$\frac{\sin 30^\circ}{\cos 30^\circ} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}$$
5. 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°.

## Simplifying trigonometric expressions using the identity

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 $\tan \theta \times \cos \theta$ as much as possible.

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

**Check your understanding**

Test your understanding of simplification

1. What is the simplified form of $\frac{\sin \theta}{\tan \theta}$?

   - cos θ
   - sin θ
   - tan θ
   - 1

   *Answer:* cos θ

   *Why:* Correct: substitute tan θ = sin θ / cos θ, so the fraction becomes sin θ divided by (sin θ / cos θ) = cos θ.

> **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.

*Calculator:* allowed

## Solving trigonometric equations using the identity

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 $2 \sin \theta = \tan \theta$ for 0° ≤ θ < 360°, where cos θ ≠ 0.

1. Step 1: Substitute tan θ with sin θ / cos θ in the equation.
2. $$2 \sin \theta = \frac{\sin \theta}{\cos \theta}$$
3. Step 2: Rearrange all terms to one side and factor out the common sin θ term.
4. $$2 \sin \theta - \frac{\sin \theta}{\cos \theta} = 0 \implies \sin \theta \left(2 - \frac{1}{\cos \theta}\right) = 0$$
5. Step 3: Set each factor equal to zero and solve for θ.
6. Case 1: sin θ = 0. Solutions in the range are θ = 0°, 180°.
7. Case 2: 2 - 1/cos θ = 0 → 1/cos θ = 2 → cos θ = 1/2. Solutions in the range are θ = 60°, 300°.
8. 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.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using the identity when cos θ = 0
  - Why it fails: Division by zero is undefined, so tan θ is undefined at these angles, making the identity invalid
  - Correct: Always check that cos θ ≠ 0 for your angle values before applying the identity
- **Wrong:** Dividing both sides of an equation by sin θ or cos θ to simplify
  - Why it fails: This removes any solutions where the divided term equals zero, leading to incomplete answers
  - Correct: Rearrange all terms to one side and factor out the common trigonometric term instead of dividing
- **Wrong:** Forgetting the identity is only valid for real angles where tan θ is defined
  - Why it fails: Applying it to angles like 90° will lead to mathematical errors
  - Correct: Exclude angles where cos θ = 0 (90°, 270° etc.) from your solution set when using this identity
- **Wrong:** Rewriting cos θ / sin θ as tan θ
  - Why it fails: 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: 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:** Memorizing the identity unnecessarily
  - Why it fails: It is provided on the formula sheet, so you waste memory space that could be used for non-provided formulae
  - Correct: Focus on learning to apply the identity correctly rather than memorizing it

## 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 |

## What's next

Now that you have mastered applying the $\tan \theta = \frac{\sin \theta}{\cos \theta}$ 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 $\sin^2 \theta + \cos^2 \theta \equiv 1$, 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.

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