Equality that does not hold in general for Wolfram Alpha

Question

Why for WolframAlpha $\frac{|\sin(\frac{x}{2})|}{|\cos(\frac{x}{2})|}=\sqrt{\frac{|(1-cos(x))|}{|(1+cos(x))|}}$ is not true $\forall x$? WolframAlpha

In my opinion the equality holds since $\sin(\frac{x}{2})=\pm\sqrt{\frac{1-\cos{x}}{2}}$ and $\cos(\frac{x}{2})=\pm\sqrt{\frac{1+\cos{x}}{2}}$