Double angles become rational functions of $\tan t$: sine gets $2u/(1+u^2)$ and cosine gets $(1-u^2)/(1+u^2)$.Suggest a slogan
For every \(t\in\RR\) with \(\cos t\neq0\), \[ \begin{aligned} \sin(2t) &= {2\tan(t) \over 1+\tan^2(t)}, \\ \cos(2t) &= {1-\tan^2(t) \over 1+\tan^2(t)}. \end{aligned} \]
How are $\sin(2t)$ and $\cos(2t)$ written in terms of $\tan(t)$?