Solving Polynomial in $\sin$ and $\cos$ Given the following equation:
$$A \sin(t) + B\cos(t) =C\sin(t)\cos(t).$$
Can we solve the above equation without the need of using general solution of 4-order polynomial equation. We can use computer but not using the search algorithm.
Please help me, this problem can from the minimization of the following function:
$$f(t) = (A + R_1\cos{t})^2 + (B+ R_2\sin{t})^2.$$
that is the problem of finding minimum distance from the origin of coordinate to a ellipse.
in which, $A$, $B$,$C$, $R_1$, $R_2$ is constant. $t$ is variable.
thank you so much.

I'm working on the same problem: 
https://math.stackexchange.com/posts/2939540/
So far I've gotten to the below equation. Here, $z_0$ is a complex number who's real and imaginary parts are functions of the semi- major and minor axes of the ellipse and the xy coordinates of the interior point. I'm not sure how to proceed further without numerical methods, but I'd think the symmetry would yield a closed form solution. 
$$z^4+2z_0z^3-2\bar{z_0}z-1=0$$
