Find zeros of this function: $$(3\tan(x)+4\cot(x))\cdot\sin(2x)$$
Do I have to multiply them and solve, or one by one, like:
$$(3\tan(x)+4\cot(x))=0$$and$$\sin(2x)=0.$$

You can proceed either way, but to assure you that you can proceed as noted secondly, we know that 
$$ab = 0 \implies a = 0 \;\text{ or }\; b = 0\;\;\text{Not necessarily both equal to zero!}$$
So
$$(3\tan(x)+4\cot(x))\cdot \sin(2x) \implies (3\tan(x)+4\cot(x))=0\;\text{ or } \sin(2x)=0$$
