solve for exact solution over interval [0*,360*)
cot(x) + 2csc(x) = 3
Multiplby by $\displaystyle \sin x$,
$\displaystyle \cot x\sin x+2\csc x\sin x=3\sin x$
Thus,
$\displaystyle \cos x+2=3\sin x$
Square both sides,
$\displaystyle (\cos x+2)^2=9\sin^2x$
$\displaystyle \cos^2 x+4\cos x+4=9\sin^2 x$
Thus,
$\displaystyle \cos^2 x+4\cos x+4=9(1-\cos^2x)$
Thus,
$\displaystyle \cos^2 x+4\cos x+4=9-9\cos^2 x$
Thus,
$\displaystyle 10\cos^2x+4\cos x-5=0$