How to solve $A\tan\theta-B\sin\theta=1$

$$\tan x=\frac{2\tan \frac x2}{1-\tan^2 \frac x2}=\frac{2t}{1-t^2}$$ $$\sin x=\frac{2\tan \frac x2}{1+\tan^2 \frac x2}=\frac{2t}{1+t^2}$$

Like any other quartic equation, there is a classical way to solve it explicitly in terms of radicals; but this is generally very messy. It is practically much easier, and no less accurate in the final analysis, to solve it by a numerical method such as Newton--Raphson.