The t-Formulae for Trigonometric Functions

We can reverse the double-angle formulae to instead construct half-angle formulae for sine and cosine. When we do this, if we perform a little bit of algebra then we can achieve a rather powerful result - both $\sin(\theta)$ and $\cos(\theta)$ can be written enitrely as a function of $\tan\left(\frac{\theta}2\right)$ . If we make the substitution $t = \tan\left(\frac{\theta}2\right)$ , then we obtain what are known as the t-formulae:

\begin{aligned} \sin(\theta) &= \frac{2t}{1 + t^2} \\ \cos(\theta) &= \frac{ 1 - t^2}{1 + t^2} \\ \end{aligned}

This is sometimes referred to as Weierstrass substitution. From these two results, we can then obtain t-formulae for $\tan(\theta)$ , $\cot(\theta)$ , $\sec(\theta)$ and $\cosec(\theta)$ . This can be very useful when we want to perform integration with complex trigonometric functions.

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