trigonometry - Can $Asin^2t + Bsin tcos t + Csin t + Dcos t + E = 0$ be solved algebraically?

This started out much more complex, but I've reduced an equation to this (it's for finding intersections of ellipses):

$$A\sin^2(t)+B\sin(t)\cos(t)+C\sin (t)+D\cos(t)+E=0$$

I want to solve for t where A/B/C/D/E are constants. Is this solvable algebraically, or is only numeric approximation possible?

Using trig identities and the formula for phase shifting, I can further simplify it down to this form:

$$\sin(2t+F) + G\sin(t+H) = I$$

Where F/G/H/I are constants. The formula is much simpler, but this may be a dead end, because now we have two angles to deal with.

Answer

yes use $$\sin(t)=2\,{\frac {\tan \left( t/2 \right) }{1+ \left( \tan \left( t/2 \right) \right) ^{2}}}$$
$$\cos(t)={\frac {1- \left( \tan \left( t/2 \right) \right) ^{2}}{1+ \left( \tan \left( t/2 \right) \right) ^{2}}}$$
and after this you can substitute $$\tan(t/2)=z$$