I have a question about why I am allowed to take the ratio of two equations to solve for a missing variable.

I am studying momentum at the moment and I have two equations.

(m1)(v1) = (m1)(v'1)cos(Ѳ1) + (m2)(v'2)cos(Ѳ2)

and

0 = (m1)(v'1)sin(Ѳ1) + (m2)(v'2)sin(Ѳ2)

I want to solve for v'2. I know how to do this by substitution using both of the equations, but my book shows another way. It says that I can solve both of them for (v'2)sin(Ѳ2) and then TAKING THE RATIO YIELDS AN EQUATION because tanѲ=sinѲ/cosѲ

so they give me tanѲ2 = (v'1sinѲ1)/((v'1cosѲ1)-v1)

Question

I have a question about why I am allowed to take the ratio of two equations to solve for a missing variable.

I am studying momentum at the moment and I have two equations.

(m1)(v1) = (m1)(v'1)cos(Ѳ1) + (m2)(v'2)cos(Ѳ2)

and

0 = (m1)(v'1)sin(Ѳ1) + (m2)(v'2)sin(Ѳ2)

I want to solve for v'2. I know how to do this by substitution using both of the equations, but my book shows another way. It says that I can solve both of them for (v'2)sin(Ѳ2) and then TAKING THE RATIO YIELDS AN EQUATION because tanѲ=sinѲ/cosѲ

so they give me tanѲ2 = (v'1sinѲ1)/((v'1cosѲ1)-v1)