Answer
The solutions to this problem is $\theta=$ {$45^\circ+n\pi, n\in Z$} or $\theta=$ {$108.4349^\circ+n\pi, n\in Z$}
Work Step by Step
$$\frac{2\tan\theta}{3-\tan^2\theta}=1$$
Since the left side is a fraction, we must build up some conditions first.
*Conditions: $$3-\tan^2\theta\ne0$$ $$\tan^2\theta\ne3$$ $$\tan\theta\ne\pm\sqrt 3$$
After building the conditions, we can solve the equation now. $$\frac{2\tan\theta}{3-\tan^2\theta}=1$$
Multiply both sides by $3-\tan^2\theta$, we have $$2\tan\theta=3-\tan^2\theta$$ $$\tan^2\theta+2\tan\theta-3=0$$ $$\tan^2\theta-\tan\theta+3\tan\theta-3=0$$ $$\tan\theta(\tan\theta-1)+3(\tan\theta-1)=0$$ $$(\tan\theta-1)(\tan\theta+3)=0$$ $$\tan\theta=1$$ or $$\tan\theta=-3$$
*For $\tan\theta=1$, $\theta=45^\circ$
A full set of values of $\theta$ therefore would be {$45^\circ+n\pi, n\in Z$}
*For $\tan\theta=-3$, $\theta=-71.565^\circ$
To eliminate the negative sign, we can add $180^\circ$ to the result, making it $\theta=108.4349^\circ$
A full set of values of $\theta$ therefore would be {$108.4349^\circ+n\pi, n\in Z$}
