Introduction
This O-Level A-Maths question tests your ability to prove a trigonometric identity by simplifying one side (usually the LHS) until it matches the RHS.
The key skills are handling fractions, using standard identities like \(\sin^2\theta + \cos^2\theta = 1\), and factorising neatly to cancel common factors.
The Question
Prove the identity:
\(\dfrac{\sin\theta}{1 – \cos\theta} – \dfrac{1}{\sin\theta} = \cot\theta\)
Step-by-Step Working (Teacher’s Method)
Step 1: Start with the LHS
\(\dfrac{\sin\theta}{1 – \cos\theta} – \dfrac{1}{\sin\theta}\)
Step 2: Use a common denominator \((1 – \cos\theta)\sin\theta\)
\(\dfrac{\sin^2\theta – (1 – \cos\theta)}{(1 – \cos\theta)\sin\theta}\)
Step 3: Expand the numerator
\(\sin^2\theta – 1 + \cos\theta\)
Step 4: Use identity \(\sin^2\theta – 1 = -\cos^2\theta\)
\(-\cos^2\theta + \cos\theta\)
Step 5: Factorise the numerator
\(\cos\theta(1 – \cos\theta)\)
Step 6: Cancel \((1 – \cos\theta)\)
\(\dfrac{\cos\theta}{\sin\theta}\)
Step 7: Convert to cotangent
\(\cot\theta\)
✅ Final Result
Identity proven ✔️
Key Concepts
| Concept | Why It Matters |
|---|---|
| \(\sin^2\theta + \cos^2\theta = 1\) | Helps rewrite \(\sin^2\theta – 1\) as \(-\cos^2\theta\) |
| Common denominator | Needed to combine fractions correctly |
| Factorisation + cancellation | Lets you simplify safely to reach the RHS |
| \(\cot\theta = \dfrac{\cos\theta}{\sin\theta}\) | Final step to match the RHS |
Tips for Students
- Always start from one side only (usually LHS)
- Use a common denominator before simplifying
- Replace \(\sin^2\theta – 1\) carefully using \(\sin^2\theta + \cos^2\theta = 1\)
- Cancel only when you have a common factor (never inside a plus/minus)
For Parents
Trigonometric proof questions build strong algebra and reasoning skills needed for many O-Level A-Maths topics (including identities, solving trig equations, and later calculus).
With enough structured practice, students learn how to simplify confidently without “guessing”.