Verified answer

tanx sinx + tanx = 0

tanx ( sinx + 1) = 0

tanx = 0

x = 0, pi, 2pi

sinx + 1 = 0

sinx = -1

x = 3pi/2

Factor.

tanx(sinx+1)=0

So,

tanx=0 or (sinx)+1=0

Since tanx=sinx/cosx, tanx=0 when sinx is 0.

And (sinx)+1=0 when sinx is -1. So just find the angles that correspond to those values of sinx.