If we write \(t = \tan \theta\), then the following equations are true.

\[\begin{align*} \tan 2\theta &= \frac{2t}{1-t^2}, \\ \sin 2\theta &= \frac{2t}{1+t^2}, \\ \cos 2\theta &= \frac{1-t^2}{1+t^2}. \end{align*}\]Can you use this diagram to obtain these formulae?

What lengths and angles might be useful to us here? Can you find the missing lengths and angles in terms of \(t\)?

Which angle is \(2\theta\)?

For what range of values of \(\theta\) does this argument work?

The relationships are shown using a triangle. Are there any values of \(\theta\) for which the diagram does not work? Can you adapt it?