5. (a) Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \cot ^ { 2 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta\).
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(a) \(\sin^2\theta + \cos^2\theta = 1\) divided by \(\sin^2\theta\): \(\frac{\sin^2\theta}{\sin^2\theta} + \frac{\cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta}\)M1
\(1 + \cot^2\theta = \operatorname{cosec}^2\theta\) cso A1
(2)
Alternative for (a): \(1 + \cot^2\theta = 1 + \frac{\cos^2\theta}{\sin^2\theta} = \frac{\sin^2\theta + \cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta} = \operatorname{cosec}^2\theta\)M1 cso A1
(b) \(2(\operatorname{cosec}^2\theta - 1) - 9\operatorname{cosec}\theta = 3\)M1
\(2\operatorname{cosec}^2\theta - 9\operatorname{cosec}\theta - 5 = 0\) or \(5\sin^2\theta + 9\sin\theta - 2 = 0\) M1
\((2\operatorname{cosec}\theta + 1)(\operatorname{cosec}\theta - 5) = 0\) or \((5\sin\theta - 1)(\sin\theta + 2) = 0\) M1
\(\operatorname{cosec}\theta = 5\) or \(\sin\theta = \frac{1}{5}\) A1
\(\theta = 11.5°, 168.5°\) A1 A1
(6) [8]
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**(a)** $\sin^2\theta + \cos^2\theta = 1$ divided by $\sin^2\theta$: $\frac{\sin^2\theta}{\sin^2\theta} + \frac{\cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta}$ | M1 |
$1 + \cot^2\theta = \operatorname{cosec}^2\theta$ | cso A1 | (2)
**Alternative for (a):** $1 + \cot^2\theta = 1 + \frac{\cos^2\theta}{\sin^2\theta} = \frac{\sin^2\theta + \cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta} = \operatorname{cosec}^2\theta$ | M1 cso A1 |
**(b)** $2(\operatorname{cosec}^2\theta - 1) - 9\operatorname{cosec}\theta = 3$ | M1 |
$2\operatorname{cosec}^2\theta - 9\operatorname{cosec}\theta - 5 = 0$ or $5\sin^2\theta + 9\sin\theta - 2 = 0$ | M1 |
$(2\operatorname{cosec}\theta + 1)(\operatorname{cosec}\theta - 5) = 0$ or $(5\sin\theta - 1)(\sin\theta + 2) = 0$ | M1 |
$\operatorname{cosec}\theta = 5$ or $\sin\theta = \frac{1}{5}$ | A1 |
$\theta = 11.5°, 168.5°$ | A1 A1 | (6) [8]
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5. (a) Given that $\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1$, show that $1 + \cot ^ { 2 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta$.\\
(b) Solve, for $0 \leqslant \theta < 180 ^ { \circ }$, the equation
$$2 \cot ^ { 2 } \theta - 9 \operatorname { cosec } \theta = 3$$
giving your answers to 1 decimal place.\\
\hfill \mbox{\textit{Edexcel C3 2008 Q5 [8]}}