Introduction
Many students lose marks in a trigo proving question because they expand too quickly or mix up identities. The good news: most O Level proofs become easy when you simplify one side carefully and use the right identity at the right time.
In this guide, we’ll break down this trigo proving question in a clean, exam-ready way.
The Question / Scenario Explanation
Source: TRIGO PROVING O Level 2025, Paper 1
Prove that
\(\frac{1 + \sin 2\theta}{\sin \theta} – \frac{1 + \cos 2\theta}{\cos \theta} = \cosec \theta\)
Step-by-Step Solution / Explanation
Step 1: Use double-angle identities
For this trigo proving question, use:
- \(\sin 2\theta = 2\sin \theta \cos \theta\)
- \(\cos 2\theta = 2\cos^2 \theta – 1\) (this form is very useful here)
Start with the LHS:
\(\frac{1 + \sin 2\theta}{\sin \theta} – \frac{1 + \cos 2\theta}{\cos \theta}\)
Substitute:
\(\frac{1 + 2\sin \theta \cos \theta}{\sin \theta} – \frac{1 + (2\cos^2 \theta – 1)}{\cos \theta}\)
Step 2: Simplify each fraction separately
First fraction:
\(\frac{1}{\sin \theta} + \frac{2\sin \theta \cos \theta}{\sin \theta} = \cosec \theta + 2\cos \theta\)
Second fraction:
\(\frac{2\cos^2 \theta}{\cos \theta} = 2\cos \theta\)
Step 3: Finish the proof
Now subtract:
\((\cosec \theta + 2\cos \theta) – (2\cos \theta) = \cosec \theta\)
So the expression equals the RHS:
\(\frac{1 + \sin 2\theta}{\sin \theta} – \frac{1 + \cos 2\theta}{\cos \theta} = \cosec \theta\)
✅ Proved. This trigo proving question simplifies neatly once the correct identity form is chosen.
Key Concepts Students Must Know
- Double-angle identity: \(\sin 2\theta = 2\sin \theta \cos \theta\)
- Useful double-angle form: \(\cos 2\theta = 2\cos^2 \theta – 1\)
- Basic conversion: \(\cosec \theta = \frac{1}{\sin \theta}\)
- Good habit for any trigo proving question: simplify one side (usually LHS) until it matches the RHS
Exam Tips / Common Mistakes
Exam Tips
- Pick the identity form that cancels nicely (here, \(\cos 2\theta = 2\cos^2 \theta – 1\) makes the second fraction clean).
- Simplify fraction-by-fraction before expanding everything.
- Always rewrite \(\frac{1}{\sin \theta}\) as \(\cosec \theta\) near the end so the RHS becomes obvious.
Common Mistakes
- Using \(\cos 2\theta = 1 – 2\sin^2 \theta\) here can make the algebra longer.
- Forgetting to split: \(\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}\) (this is what reveals \(\cosec \theta\)).
- Cancelling incorrectly: \(\frac{2\sin \theta \cos \theta}{\sin \theta} = 2\cos \theta\) (only \(\sin \theta\) cancels).
Parent Insight
Trigonometry proofs look intimidating, but they are pattern-based. When students practise 10–20 proofs with the same structure, they start recognising which identities to use quickly. That speed and confidence matters a lot in O Level Paper 1.
Conclusion
This trigo proving question is best solved by substituting \(\sin 2\theta\) and using the smart form \(\cos 2\theta = 2\cos^2 \theta – 1\). After splitting and cancelling carefully, the expression becomes \(\cosec \theta\), which matches the RHS exactly.
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