Introduction
This question focuses on proving a trigonometric identity, a key skill tested frequently in O-Level A-Maths.
To succeed, students must:
- convert reciprocal identities correctly
- simplify step by step
- avoid cancelling terms incorrectly
We follow the teacher’s exact board working, step by step.
The Question
Prove that:
\(\dfrac{1}{(1 + \csc\theta)(\sec\theta – \tan\theta)} = \tan\theta\)
Step-by-Step Working (Teacher’s Method)
Step 1: Start with the left-hand side
\(\dfrac{1}{(1 + \csc\theta)(\sec\theta – \tan\theta)}\)
Step 2: Rewrite using reciprocal identities
\(\csc\theta = \dfrac{1}{\sin\theta},\quad \sec\theta = \dfrac{1}{\cos\theta},\quad \tan\theta = \dfrac{\sin\theta}{\cos\theta}\)
Step 3: Substitute into the expression
\(\dfrac{1}{\left(1 + \dfrac{1}{\sin\theta}\right)\left(\dfrac{1}{\cos\theta} – \dfrac{\sin\theta}{\cos\theta}\right)}\)
Step 4: Simplify each bracket
\(\dfrac{1}{\dfrac{1 + \sin\theta}{\sin\theta} \cdot \dfrac{1 – \sin\theta}{\cos\theta}}\)
Step 5: Flip and multiply
\(= \dfrac{\sin\theta}{1 + \sin\theta} \cdot \dfrac{\cos\theta}{1 – \sin\theta}\)
Step 6: Multiply numerators and denominators
\(= \dfrac{\sin\theta \cos\theta}{(1 + \sin\theta)(1 – \sin\theta)}\)
Step 7: Use identity \((1 + \sin\theta)(1 – \sin\theta) = 1 – \sin^2\theta\)
\(= \dfrac{\sin\theta \cos\theta}{1 – \sin^2\theta}\)
Step 8: Apply Pythagorean identity
\(1 – \sin^2\theta = \cos^2\theta\)
Step 9: Final simplification
\(\dfrac{\sin\theta \cos\theta}{\cos^2\theta} = \dfrac{\sin\theta}{\cos\theta} = \tan\theta\)
✅ Final Result
\(\tan\theta\)
Identity proven ✔️
Key Concepts
| Concept | Why It Matters |
|---|---|
| Reciprocal identities | Converts expressions into solvable form |
| Algebraic simplification | Prevents cancellation errors |
| Pythagorean identity | Essential for trig proofs |
| Step-by-step logic | Required for full method marks |
Tips for Students
- Always start from one side only
- Convert sec, tan, cosec into sine & cosine
- Simplify brackets before cancelling
- Use identities confidently and aim to reach the RHS exactly
For Parents
Trigonometric identity proofs strengthen:
- logical thinking
- algebra fluency
- exam confidence
They are a core component of O-Level A-Maths.