**(i)**

$(\sin 22.5 + \cos 22.5)^2 = \sin^2 22.5 + \cos^2 22.5 + ......$ | M1 | Attempts to expand $(\sin 22.5 + \cos 22.5)^2$. Award if you see $\sin^2 22.5 + \cos^2 22.5 + ......$ There must be $> $ two terms. Condone missing brackets ie $\sin 22.5^2 + \cos 22.5^2 + ......$ |

$= \sin^2 22.5 + \cos^2 22.5 + 2\sin 22.5 \cos 22.5$ | B1 | Stating or using $\sin^2 22.5 + \cos^2 22.5 = 1$. Accept $\sin 22.5^2 + \cos 22.5^2 = 1$ as the intention is clear. Note that this may also come from using the double angle formula $\sin^2 22.5 + \cos^2 22.5 = (\frac{1 - \cos 45}{2}) + (\frac{1 + \cos 45}{2}) = 1$ |

Uses $2\sin x \cos x = \sin 2x \Rightarrow 2\sin 22.5 \cos 22.5 = \sin 45$ | M1 | Uses $2\sin x \cos x = \sin 2x$ to write $2\sin 22.5 \cos 22.5$ as $\sin 45$ or $\sin(2 \times 22.5)$ |

$(\sin 22.5 + \cos 22.5)^2 = 1 + \sin 45$ | A1 | Reaching the intermediate answer $1 + \sin 45$ |

$= 1 + \frac{\sqrt{2}}{2}$ or $1 + \frac{1}{\sqrt{2}}$ | cso A1 | $\text{Cso} 1 + \frac{\sqrt{2}}{2}$ or $1 + \frac{1}{\sqrt{2}}$. Be aware that both $1.707$ and $\frac{2 + \sqrt{2}}{2}$ can be found by using a calculator for $1 + \sin 45$. Neither can be accepted on their own without firstly seeing one of the two answers given above. **Each stage should be shown as required by the mark scheme.** Note that if the question states use $(\sin \theta + \cos \theta)^2$ they can pick up the first M and B marks, but no others until the use $\theta = 22.5$. All other marks then become available. | (5 marks)

**(ii)(a)**

$\cos 2\theta + \sin \theta = 1 \Rightarrow 1 - 2\sin^2 \theta + \sin \theta = 1$ | M1 | Substitutes $\cos 2\theta = 1 - 2\sin^2 \theta$ in $\cos 2\theta + \sin \theta = 1$ to produce an equation in $\sin \theta$ only. It is acceptable to use $\cos 2\theta = 2\cos^2 \theta - 1$ or $\cos^2 \theta - \sin^2 \theta$ as long as the $\cos^2 \theta$ is subsequently replaced by $1 - \sin^2 \theta$. |

$\sin \theta - 2\sin^2 \theta = 0$ or $k = 2$ | A1* | Obtains the correct simplified equation in $\sin \theta$. $\sin \theta - 2\sin^2 \theta = 0$ or $\sin \theta = 2\sin^2 \theta$ must be written in the form $2\sin^2 \theta - \sin \theta = 0$ as required by the question. Also accept $k = 2$ as long as no incorrect working is seen. | (2 marks)

**(ii)(b)**

$\sin \theta(2\sin \theta - 1) = 0$ | M1 | Factorises or divides by $\sin \theta$. For this mark $1 = 'k' \sin \theta$ is acceptable. If they have a 3 TQ in $\sin \theta$ this can be scored for correct factorisation. |

$\sin \theta = 0$, $\sin \theta = \frac{1}{2}$ | A1 | **Both** $\sin \theta = 0$, **and** $\sin \theta = \frac{1}{2}$ |

Any two answers from $0, 30, 150, 180$. | B1 | Any two answers from $0, 30, 150, 180$. |

All four answers $0, 30, 150, 180$ with no extra solutions inside the range. Ignore solutions outside the range. | A1 | All four answers $0, 30, 150, 180$ with no extra solutions inside the range. Ignore solutions outside the range. | (4 marks)

| (11 marks) |

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# Question 6.alt 1:

**(i)**

$(\sin 22.5 + \cos 22.5)^2 = \sin^2 22.5 + \cos^2 22.5 + ......$ | M1 | |

$= \sin^2 22.5 + \cos^2 22.5 + 2\sin 22.5 \cos 22.5$ | B1 | |

Uses $2\sin x \cos x = 2, \sqrt{1 - \cos 2x}, \sqrt{\cos 2x + 1} \Rightarrow \sqrt{1 - \cos 45} \sqrt{1 + \cos 45}$ | M1 | |

$= \sqrt{1 - \cos^2 45}$ | A1 | |

Hence $(\sin 22.5 + \cos 22.5)^2 = 1 + \frac{\sqrt{2}}{2}$ or $1 + \frac{1}{\sqrt{2}}$ | A1 | | (5 marks)

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# Question 6.alt 2:

**(i)**

Uses Factor Formula $(\sin 22.5 + \sin 67.5)^2 = (2 \sin 45 \cos 22.5)^2$ | M1, A1 | |

Reaching the stage $= 2\cos^2 22.5$ | B1 | |

Uses the double angle formula $= 2\cos^2 22.5 = 1 + \cos 45$ | M1 | |

$= 1 + \frac{\sqrt{2}}{2}$ or $1 + \frac{1}{\sqrt{2}}$ | A1 | | (5 marks)

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# Question 6.alt 3:

**(i)**

Uses Factor Formula $(\cos 67.5 + \cos 22.5)^2 = (2 \cos 45 \cos 22.5)^2$ | M1, A1 | |

Reaching the stage $= 2\cos^2 22.5$ | B1 | |

Uses the double angle formula $= 2\cos^2 22.5 = 1 + \cos 45$ | M1 | |

$= 1 + \frac{\sqrt{2}}{2}$ or $1 + \frac{1}{\sqrt{2}}$ | A1 | | (5 marks)

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