Answer:
[tex]\large\boxed{\theta=-\dfrac{\pi}{4}+k\pi\ or\ \theta=k\pi}\ for\ k\in\mathbb{Z}[/tex]
Step-by-step explanation:
[tex]\tan^2\theta+\tan\theta=0\\\\\tan\theta(\tan\theta+1)=0\iff\tan\theta=0\ \vee\ \tan\theta+1=0\\\\\tan\theta=0\Rightarrow \theta=k\pi\ \text{for}\ k\in\mathbb{Z}\\\\\tan\theta+1=0\qquad\text{subtract 1 from both sides}\\\\\tan\theta=-1\Rightarrow\theta=-\dfrac{\pi}{4}+k\pi\ \text{for}\ k\in\mathbb{Z}[/tex]
