I don't remember where I saw this equation but it was difficult to solve. The equation was:
$$\frac{\tan(x)}{1+\sin(x)} = q$$
where $q$ is a constant. When I started to solve this equation it made me baffled, so I am asking this question to check whether there are any alternative methods, rather than doing simple calculations.
Note that $\tan(x) = \sin(x)/\cos(x)$, and $\cos^2(x) + \sin^2(x) = 1$.
Henceforth, I just use $t, s, c$ as shorthand.
Your question is to solve for $q$ given $\frac{t}{1 + s} = q$. Observe:
$$\frac{t}{1+s} = \frac{s}{(1+s)c} = \frac{s}{(1+s)\sqrt{1 - s^2}}$$
At this point, you have an expression in one variable. And so setting it equal to a given number $q$ yields an equation in one variable, which you can then solve.