Prove or show algebraic identity

7. (a) Show that $$12 \sin ^ { 2 } x - \cos x - 11 = 0$$ may be expressed in the form $$12 \cos ^ { 2 } x + \cos x - 1 = 0$$ (b) Hence, using trigonometry, find all the solutions in the interval \(0 \leqslant x \leqslant 360 ^ { \circ }\) of $$12 \sin ^ { 2 } x - \cos x - 11 = 0$$ Give each solution, in degrees, to 1 decimal place.
\includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-15_106_97_2615_1784}

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { 2 } ( \tan x + \cot x ) \quad \frac { \pi } { 6 } \leqslant x \leqslant \frac { \pi } { 3 }$$

1. Show that the length of \(C\) is given by $$\frac { 1 } { 2 } \int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 3 } } \left( \tan ^ { 2 } x + \cot ^ { 2 } x \right) d x$$
2. Hence determine the exact length of \(C\), giving your answer in simplest form.

2 One of the equations below is true for all values of \(x\)
Identify the correct equation.
Tick \(( \checkmark )\) one box.
\(\cos ^ { 2 } x = - 1 - \sin ^ { 2 } x\) □
\(\cos ^ { 2 } x = - 1 + \sin ^ { 2 } x\)
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-02_113_113_1658_790}
\(\cos ^ { 2 } x = 1 - \sin ^ { 2 } x\)
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-02_113_113_1813_790}
\(\cos ^ { 2 } x = 1 + \sin ^ { 2 } x\) □