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These identities supply the formulas used in the tangent half-angle substitution. The half-angle identities include

The tangent half-angle substitution $t=\tan (x/2)$ converts rational trigonometric expressions into rational functions: $\sin x=2t/(1+t^2)$, $\cos x=(1-t^2)/(1+t^2)$ and $dx=2dt/(1+t^2)$.