How did they "divide the equations"?

1. The problem statement, all variables and given/known data



Two 5.0g point charges on 1.0m-long threads repel each other after being charged to +100nC.

What is angle theta? You can assume theta is a small angle.







2. Relevant equations

[itex]K=9.0*10^9 Nm^2/C^2[/itex]

[itex]g=9.8m/s^2[/itex]

[itex]q=\frac{K|q_1||q_2|}{d^2}[/itex]



3. The attempt at a solution



I determined my unknowns and translated everything into standard units, and drew out the free-body diagram.







Translated everything into standard units.

5.0g = 5.0*10^(-3) kg

100nc = 100*10^(-9) C



[itex]\frac{1}{2}d=\sin{\theta}[/itex]

[itex]d=2\sin{\theta}[/itex]

[itex]T_x=T\sin{\theta}=F_{2on1}=\frac{Kq^2}{d^2}[/itex]

[itex]T_x=T\sin{\theta}=F_{2on1}=\frac{Kq^2}{(2\sin{\theta})^2}[/itex]

[itex]T_x=T\sin{\theta}=F_{2on1}=\frac{Kq^2}{4sin^2{\theta} }[/itex]

[itex]T_y=T\cos{\theta}=mg[/itex]



The "missing" step:

The solutions manual says that the next step from here is to "divide the two equations and solve for q". What I don't know is how they "divided the two equations" to get this:



[itex]\sin^2{\theta}\tan{\theta}=\frac{Kq^2}{4L^2mg}=4.59*10^{-4}[/itex]