# OCR A Level: Core 3 - Trigonometry [January 2013 Q9]

Geometry Level 4

$$(\text{i})$$ Prove that $\cos^2{(\theta + 45°)}-\dfrac{1}{2}(\cos 2 \theta - \sin 2 \theta) \equiv \sin^2 \theta .$

$$(\text{ii})$$ Hence solve $6\cos^2{ \left ( \dfrac{\theta}{2} + 45° \right )}-3(\cos \theta - \sin \theta) = 2$ for $$-90°< \theta < 90°$$.

$$(\text{iii})$$ It is given that there are two values of $$\theta$$, where $$-90°< \theta < 90°$$, satisfying $6\cos^2{\left (\frac{\theta}{3} + 45°\right )}-3\left (\cos \frac{2 \theta}{3} - \sin \dfrac{2 \theta}{3} \right ) = k.$

Find the set of possible values for $$k$$.

If $$a<k<\dfrac{b}{c}$$ where $$a$$, $$b$$ and $$c$$ are integers with $$b$$ and $$c$$ coprime, input $$a+b+c$$ as your answer.

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