I have a system of equations as the following:
$$ \left\{\begin{array}{rcl} f_1(x,y)=0\\ \frac{f_2(x,y)}{g_1(x)}=0\\ f_3(x,y)=0\\ \frac{f_4(x,y)}{g_2(x)}=0 \end{array} \right. $$
where $g_1(x)$ and $g_2(x)$ are functions that contain trigonometric functions like $\cos x$ or $\cos^2 x$.
I know that I could multiply each line by a different scalar without affecting the solution. However, how will the solution be affected if I multiply the second line by $g_1(x)$ and the third line by $g_2(x)$ in order to eliminate these terms from the denominator?
Short answer - yes, if you're careful.
Longer answer:
A fraction is $0$ if and only if its numerator is $0$, so you can multiply your fractions by the functions in their denominators as long as those denominators are not $0$. When the functions in the denominator are $0$ the fractions are ambiguous - they may make sense if the numerator vanishes too in such a way as to cause the fraction to have a limit.