Evaluate the integral... and maybe some trig identities

Relevant wiki: Half Angle Tangent Substitution

$\begin{aligned} A & = \int_0^{4\pi} \frac {2\tan \frac x2}{\tan^2 \frac x2 + 1}\ dx & \small \color{#3D99F6} \text{See half-angle tangent substitution.} \\ & = \int_0^{4\pi} \sin x \ dx \\ & = 2\int_0^{2\pi} \sin x \ dx \\ & = 2 \int_{-\pi}^\pi \sin x \ dx & \small \color{#3D99F6} \text{Note that }\sin x \text{ is an odd function.} \\ & = \boxed{0} \end{aligned}$