7. (a) Prove that $$\frac{\sin \theta \cdot \cos \theta}{\cos^2 \theta} + \frac{\sin^2 \theta}{\sin \theta} = 2 \csc 2\theta, \quad \theta \neq 90^\circ.$$ (b) On the axes on page 20, sketch the graph of $$y = 2 \csc 2\theta$$ for $$0^\circ < \theta < 360^\circ.$$ (c) Solve, for $$0^\circ < \theta < 360^\circ$$, the equation $$\frac{\sin \theta \cdot \cos \theta}{\cos^2 \theta} \cdot \frac{\cos \theta}{\sin \theta} = 3,$$ giving your answers to 1 decimal place. - Edexcel - A-Level Maths Pure - Question 1 - 2007 - Paper 5

To sketch the graph:

1. Identify the asymptotes where $2\theta = n\pi$ for $n \in \mathbb{Z}$, which gives asymptotes at: $\theta = 0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ$.

2. Mark the points where the function attains its maximum and minimum, which occur at $\theta = 45^\circ, 135^\circ, 225^\circ, 315^\circ$ resulting in values of 2 and -2.

3. Plot the curve, noting the general shape of the cosecant function - it will have branches extending infinitely in both positive and negative directions.

To solve the equation:

1. Rearrange and simplify the equation: $2\csc 2\theta = 3$.

2. Therefore, $\csc 2\theta = \frac{3}{2}$ implies: $\sin 2\theta = \frac{2}{3}$.

3. Find the general solution for $2\theta$: $2\theta = \sin^{-1}\left(\frac{2}{3}\right), 180^\circ - \sin^{-1}\left(\frac{2}{3}\right)$ which gives: $2\theta \approx 41.81^\circ, 138.19^\circ, 401.81^\circ, 498.19^\circ$.

4. Hence, for $\theta$, divide each result by 2: $\theta \approx 20.90^\circ, 69.10^\circ, 200.90^\circ, 249.10^\circ.$
   Format the answers to 1 decimal place as:

   • $20.9^\circ$, $69.1^\circ$, $200.9^\circ$, $249.1^\circ$.