Use \(\cot2x=(\cot^2x-1)/(2\cot x)\), where the original functions are defined. Then \[\frac{\cot^2x-1}{\cot x}+3\cot x=5.\] Let \(u=\cot x\). Multiplying by \(u\), \[4u^2-5u-1=0,\qquad u=\frac{5\pm\sqrt{41}}8.\] Equivalently, \[\tan x=\frac{-5\pm\sqrt{41}}2.\] The positive tangent root gives the quadrant I angle, and the negative root gives the quadrant II angle. Thus \[\boxed{x\approx35.1^\circ,\quad99.9^\circ}.\] Neither root is an excluded cotangent point.