Clear the denominator: \[\tan\theta+2\sin\theta=3\tan\theta-6\sin\theta,\qquad2\tan\theta-8\sin\theta=0.\] For the defined part of the interval, multiply by \(\cos\theta\): \[2\sin\theta(1-4\cos\theta)=0.\] The roots of \(\sin\theta=0\) are excluded endpoints. Thus \(\cos\theta=1/4\), giving \[\boxed{\theta=75.5^\circ}.\] This value satisfies the original equation; \(90^\circ\) is undefined.