Functions of the form $\frac{1}{\sin x}$ occur so frequently that we give them their own names:

Starting with the identity ${\sin }^{2}x+{\cos }^{2}x=1$, we can show that

and

1. Sketch the curve

   $$y=\text{cosec}x,-2\pi \le x\le 2\pi$$

2. Show that

   $$1+{\cot }^{2}x={\text{cosec}}^{2}x$$

3. Solve the equation

   $$4{\cot }^2\theta +7=8\text{cosec}\theta$$

   giving all $\theta$ satisfying $0\le \theta \le 2\pi$