Just want to get this little crease in my trig knowledge, ironed out. So, a few months ago, I made the classic newbie mistake. I had a nasty trig equation which I needed to simplify, in order to solve for theta in a given interval. In doing so, I divided by a trig function and consequently I lost a few solutions. When my teacher noticed what I did, he said never divide something by a trig function instead expand the bracket or subtract. When he told me this instead of asking why this was the case, I just placed it into memory and continued striving to become #1 in the class. If I remember correctly, the trig equation I had to solve was: $$ \tan(\theta)=\tan(\theta)(2+3\sin(\theta)) $$ $$ 0\leq\theta\leq360 $$
The immediate step I took was to divide both sides by $ \tan(\theta)$, resulting in 1. When I try to think about why we lose solutions, the following comes to mind: Well, I know division by 0 is undefined and in effect that's what I could be doing since the domain has several spots where $ \tan(\theta)=0 $. However, I still feel this doesn't fully justify why we may lose some solutions and retain some. Please forgive any grammar mistakes.