Weierstrass Substitution/Derivative

Proof Technique
Let:
 * $u \leftrightarrow \tan \dfrac \theta 2$

for $-\pi < \theta < \pi$, $u \in \R$.

Then:
 * $\dfrac {\d \theta} {\d u} = \dfrac 2 {1 + u^2}$

Proof
Let $u = \tan \dfrac \theta 2$ for $-\pi < \theta < \pi$.

From Shape of Tangent Function, this substitution is valid for all real $u$.

Then:

Also see

 * Hyperbolic Tangent Half-Angle Substitution