8. Solve
$$\operatorname { cosec } ^ { 2 } 2 x - \cot 2 x = 1$$
for \(0 \leqslant x \leqslant 180 ^ { \circ }\).
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Question 8: \(\text{cosec}^2 2x - \cot 2x = 1,\quad 0 \leq x \leq 180°\)
Answer Marks
Guidance
Answer/Working Mark
Guidance
Using \(\text{cosec}^2 2x = 1 + \cot^2 2x\) gives \(1 + \cot^2 2x - \cot 2x = 1\) M1
Writing/using \(\text{cosec}^2 2x = \pm 1 \pm \cot^2 2x\)
\(\cot^2 2x - \cot 2x = 0\) or \(\cot^2 2x = \cot 2x\) A1
\(\cot 2x(\cot 2x - 1) = 0\) or \(\cot 2x = 1\) dM1
Attempt to factorise or solve quadratic, or cancel \(\cot 2x\)
\(\cot 2x = 0\) and \(\cot 2x = 1\) A1
\(\cot 2x = 0 \Rightarrow \tan 2x \to \infty \Rightarrow 2x = 90°, 270° \Rightarrow x = 45°, 135°\) ddM1
Candidate attempts to divide principal angles by 2
\(\cot 2x = 1 \Rightarrow \tan 2x = 1 \Rightarrow 2x = 45°, 225° \Rightarrow x = 22.5°, 112.5°\) \(x = \{22.5°, 45°, 112.5°, 135°\}\) A1, B1
Both \(x=22.5\) and \(x=112.5\); Both \(x=45\) and \(x=135\)
> Note: Extra solutions inside \(0 \leq x \leq 180°\) causing otherwise full marks: withhold final accuracy mark.
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## Question 8: $\text{cosec}^2 2x - \cot 2x = 1,\quad 0 \leq x \leq 180°$
| Answer/Working | Mark | Guidance |
|---|---|---|
| Using $\text{cosec}^2 2x = 1 + \cot^2 2x$ gives $1 + \cot^2 2x - \cot 2x = 1$ | M1 | Writing/using $\text{cosec}^2 2x = \pm 1 \pm \cot^2 2x$ |
| $\cot^2 2x - \cot 2x = 0$ or $\cot^2 2x = \cot 2x$ | A1 | |
| $\cot 2x(\cot 2x - 1) = 0$ or $\cot 2x = 1$ | dM1 | Attempt to factorise or solve quadratic, or cancel $\cot 2x$ |
| $\cot 2x = 0$ and $\cot 2x = 1$ | A1 | |
| $\cot 2x = 0 \Rightarrow \tan 2x \to \infty \Rightarrow 2x = 90°, 270° \Rightarrow x = 45°, 135°$ | ddM1 | Candidate attempts to divide principal angles by 2 |
| $\cot 2x = 1 \Rightarrow \tan 2x = 1 \Rightarrow 2x = 45°, 225° \Rightarrow x = 22.5°, 112.5°$ | | |
| $x = \{22.5°, 45°, 112.5°, 135°\}$ | A1, B1 | Both $x=22.5$ and $x=112.5$; Both $x=45$ and $x=135$ |
> **Note:** Extra solutions inside $0 \leq x \leq 180°$ causing otherwise full marks: withhold final accuracy mark.
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8. Solve
$$\operatorname { cosec } ^ { 2 } 2 x - \cot 2 x = 1$$
for $0 \leqslant x \leqslant 180 ^ { \circ }$.\\
\hfill \mbox{\textit{Edexcel C3 2010 Q8 [7]}}