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

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