The general solution of dydx=2ytan⁡x+tan2⁡x is

dydx=2ysinxcosx+sin2xcos2x cos2⁡xdydx−ysin2x=sin2⁡x→dcos2⁡x⋅y=sin2⁡x=12(1−cos⁡2x) ∫dcos2⁡x⋅y=12∫1dx−∫cos⁡2x dx⇒ycos2x=x2−14sin⁡2x+c