Is $\sin(\alpha)=\frac{\tan(\alpha)}{\sqrt{1+\tan^2(\alpha)}}$ a true statment?

If the domain is $-\pi/2 < \alpha < \pi/2$, then the two functions are identical. However, if the domain contains any values for which $\cos(\alpha) < 0$, then as you've shown the functions differ by a factor of $-1$.

You can see it on this graph - the functions track together, then the second one suddenly flips, then flips back, and so on.

For angles $0 < \alpha < \pi/2 $ here is a geometric proof. The pictured right-angled triangle exists. Now what is the sine of $\alpha$ in that picture?