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Simplify \(\sin \, \theta \, \cos \theta \,\operatorname{cosec}^2 \, \theta\). (2 marks) --- 4 WORK AREA LINES (style=lined) ---
\(\cot \theta\)
\(\sin \theta \cos \theta \times \dfrac{1}{\sin ^2 \theta}\)
\(=\dfrac{\cos \, \theta}{\sin \, \theta}\)
\(=\cot \, \theta\)
Express \(3 \operatorname{cosec}(180+x)+5 \cos (90-x)\) as a single fraction in terms of \(\sin x\), given all angles are measured in degrees. (3 marks) --- 8 WORK AREA LINES (style=lined) --- \(\dfrac{-3+5 \sin ^2 x}{\sin x}\) \(3 \operatorname{cosec}(180+x)+5 \cos (90-x)\) \(=\dfrac{3}{\sin \left(180^{\circ}+x\right)}+5 \sin x\) \(=\dfrac{3}{-\sin x}+5 \sin x\) \(=\dfrac{-3}{\sin x}+\dfrac{5 \sin ^2 x}{\sin x}\) \(=\dfrac{-3+5 \sin ^2 x}{\sin x}\)
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