The t-formulae

The t-substitution

With $t=\tan\frac x2$: $\sin x=\frac{2t}{1+t^2}$, $\cos x=\frac{1-t^2}{1+t^2}$, $\tan x=\frac{2t}{1-t^2}$, and $dx=\frac{2\,dt}{1+t^2}$. These turn any rational expression in $\sin x$ and $\cos x$ into an algebraic one in $t$, for solving equations and integrating.