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?

Right angled triangle containing the angle theta and with base length 2. A line is drawn from the top corner of the right anlged triangle to the base, forming an isoceles triangle with the equal angles being theta

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?