\tan\frac{x}{2}

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1.26 Tangent substitutions or $\tan\frac{x}{2}$ substitutions

The subsitution $t=\tan\frac{x}{2}$ converts a rational function of $\sin x$ and $\cos x$ into a rational function of $t$ .

Proof: We claim that

{{dx}\over{dt}}={{2}\over{1+t^{2}}},\quad\sin x={{2t}\over{1+t^{2}}},\quad\cos x% ={{1-t^{2}}\over{1+t^{2}}}.

For the first equality, we differentiate

{{{dt}\over{dx}}}\,{={{1}\over{2}}\sec^{2}\frac{x}{2}}\,{={{1}\over{2}}\bigl(1% +\tan^{2}\frac{x}{2}\bigr)}\,{={{1}\over{2}}\bigl(1+t^{2}\bigr);}

1.26 Tangent substitutions or tan⁡x2\tan\frac{x}{2} substitutions