Standard +0.3 This is a straightforward identity proof requiring manipulation of reciprocal trig functions and application of the double angle formula. Students need to express cot and tan in terms of sin and cos, combine fractions, and recognize the double angle form—standard techniques for this topic with no novel insight required.
Express LHS in terms of \(\cos\theta\) and \(\sin\theta\)
M1
Make sufficient relevant use of double-angle formula(e)
M1
Complete proof of the result
A1
OR route:
Answer
Marks
Guidance
Express RHS in terms of \(\cos\theta\) and \(\sin\theta\) or in terms of \(\tan\theta\)
M1
Express RHS as the difference (or sum) of two fractions
M1
Complete proof of the result
A1
Guidance: SR: an attempt ending with \(\frac{1-\tan^2\theta}{\tan\theta} = \cot\theta - \tan\theta\) earns M1 B1 only
Total: 3 marks
**EITHER route:**
Express LHS in terms of $\cos\theta$ and $\sin\theta$ | M1 |
Make sufficient relevant use of double-angle formula(e) | M1 |
Complete proof of the result | A1 |
**OR route:**
Express RHS in terms of $\cos\theta$ and $\sin\theta$ or in terms of $\tan\theta$ | M1 |
Express RHS as the difference (or sum) of two fractions | M1 |
Complete proof of the result | A1 |
**Guidance:** SR: an attempt ending with $\frac{1-\tan^2\theta}{\tan\theta} = \cot\theta - \tan\theta$ earns M1 B1 only | | **Total: 3 marks**
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