(i) Express $2\text{sin}x-2\text{cos}x$ in the form $R\text{sin}(x- \alpha ^{\circ})$, where $R>0$ and $0^{\circ}< \alpha <90^{\circ}$.

Give the exact value of $R$ and the value of $\alpha$.

(ii) Hence sketch the curve with equation

$y=2(\text{sin}x-\text{cos}x)0^{\circ}\leqx\leq360^{\circ}$

Show on your sketch the coordinates of the points where the curve crosses the coordinate axes, and state the exact coordinates of the maximum and minimum turning points.

Hence show that

$\text{sin}165^{\circ}=\frac{\sqrt{6}-\sqrt{2}}{4}$

The first three terms of an arithmetic sequence are

$cosxcosx+sinxcosx+2sinxx\neqn \pi$

Given that ${S}_{9}$ represents the sum of the first 9 terms of this sequence as $x$ varies,

(i) find the exact maximum value of ${S}_{9}$

(ii) deduce the smallest positive value of $x$ at which this maximum value of ${S}_{9}$ occurs.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Show that

$\frac{5\text{sin}2x}{\text{tan}x}\equiv10{\text{cos}}^{2}xx\neq\frac{k \pi }{2}$