How do I solve this?

Question

tan^2x= 2tanxsinx

My work so far:
tan^2x - 2tanxsinx=0
tanx(tanx - 2sinx)=0

Then the solutions are:

TanX=0

and
sinX/cosX = 2 sin X

Divide through by sinX: we have to check this later to see if allowed (ie sinX is not zero)
cosX=1/2
Now, if CosX=1/2, then sinX must equal sqrt3/2, from cos^2 + sin^2=1 for any angle.

So, solutions: x = arctan 0
and x=arccos 1/2

check that.

Answer 1

tan(arccos 1/2) = sqrt3/2

and

tan(arctan 0) = 0

So, the solutions are:
x=arctan 0
and
x=arccos 1/2