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1. Find the exact values of:
  1. $\sec 225^{\circ}$
  2. $\cot \dfrac{4\pi}{3}$
2. Sketch $$y = \sec x$$ in the interval $[0, 2\pi]$, labelling the coordinates of any intersections with the axes and the turning points.
3. Sketch $$y = \cot x$$ in the interval $[0, 2\pi]$, labelling the coordinates of any intersections with the axes.
4. Sketch the graph of $$y = -\cot 2x$$ in the interval $[0, 2\pi]$, labelling the coordinates of any intersections with the axes.
5. Sketch the graph of $$y = 2\sec\left(x-\frac{\pi}{3}\right)$$ in the interval $[0, 2\pi]$, labelling the coordinates of any intersections with the axes and the turning points.
6. Sketch, in the interval $-2\pi \leq x \leq 2\pi$ $$y=3+5\cosec x$$ and write down the range of values for $k$ for which the equation $3+5\cosec x=k$ has zero solutions.
7. Prove the following:
  1. $\cot\theta+\tan\theta \equiv \cosec\theta\sec\theta$
  2. $(1-\cos x)(1+\sec x) \equiv \sin x\tan x$
8. Prove the following:
  1. $\dfrac{\cos x}{1-\sin x}+\dfrac{1-\sin x}{\cos x} \equiv 2\sec x$
  2. $\dfrac{\cos\theta}{1+\cot\theta} \equiv \dfrac{\sin\theta}{1+\tan\theta}$
9. Solve, in the interval $0\leqslant \theta \leqslant 360^{\circ}$: $$5\cot\theta=-2$$
10. Solve, in the interval $0\leqslant \theta \leqslant 360^{\circ}$: $$3\sec^2\theta-4=0$$
11. Solve, in the interval $0\leqslant \theta \leqslant 360^{\circ}$: $$\cot^2\theta-8\tan\theta=0$$
12. Solve, in the interval $0\leqslant \theta \leqslant 360^{\circ}$: $$2\sin\theta=\cosec\theta$$
13. Solve, in the interval $-\pi \leqslant \theta \leqslant \pi$: $$2\cosec^2\theta-3\cosec\theta = 0$$
14. Solve, in the interval $-\pi \leqslant \theta \leqslant \pi$: $$3\cot\theta=2\sin\theta$$
15. Prove that $$\dfrac{\sin x\tan x}{1-\cos x} - 1 \equiv \sec x$$ and hence explain why $$\dfrac{\sin x\tan x}{1-\cos x} - 1 =-0.5$$ has no solutions.
16. Given $$3\tan^2\theta+4\sec^2\theta=5$$ and that $\theta$ is obtuse, find the exact value of $\sin\theta$.
17. Given $a = 4\sec x$, $b = \cos x$, and $c=\cot x$,
  1. Find $b$ in terms of $a$.
  2. Find $c^2$ in terms of $a$.
18. Prove:
  1. $\sec^4\theta-\tan^4\theta \equiv \sec^2\theta+\tan^2\theta$
  2. $1-\cos^2\theta \equiv (\sec^2\theta-1)(1-\sin^2\theta)$
19. Prove:
  1. $\sec^2\theta+\cosec^2\theta \equiv \sec^2\theta\cosec^2\theta$
  2. $(\sec\theta-\sin\theta)(\sec\theta+\sin\theta) \equiv \tan^2\theta+\cos^2\theta$
20. Given that $x=\sec\theta + \tan\theta$,
  1. Show that $\dfrac{1}{x} = \sec\theta - \tan\theta$.
  2. Express $x^2+x^{-2}+2$ in terms of $\theta$ in its simplest form.
21. Given $$2\sec^2\theta - \tan^2\theta = p$$ write $\cosec^2\theta$ in terms of $p$.
22. Prove: $$\dfrac{\sec\theta-1}{\sec\theta+1}\equiv \tan^2\dfrac{\theta}{2}$$
23. Prove: $$\cot(A+B) \equiv \dfrac{\cot A\cot B -1}{\cot A + \cot B}$$
24. Solve, in the interval $-\pi \leqslant x \leqslant \pi$: $$\dfrac{1+\cot x}{1+\tan x}=5$$