What 's wrong with going from $x = \sin^{-1}\frac{a-b}{\sqrt{(a-b)^2+(c-d)^2}}$ to $\sin x \;\sqrt{(a-b)^2+(c-d)^2}=a-b$?

Question

I got this equation by using the Law of cosines.

$$x = \sin^{-1}\left(\frac{a-b}{\sqrt{(a-b)^2+(c-d)^2}}\right) \tag{1}$$

For my use case, I need to remove all divisions in that formula. I did so by applying sin to dissolve arcsin and then multiplying with the lower half of the fraction. My result is this equation:

$$\sin x \;\sqrt{(a-b)^2+(c-d)^2} = a-b \tag{2}$$

Apparently this is wrong, because solving it for test values results in different results.

What did I do wrong here?