I have a question about my teacher's solution to this problem:
If $\tan y = \dfrac{x}{b}$ and $\tan z = \dfrac{10+x}{b}$, solve for $y$ in terms of $x$ and $z$.
My Pre-Calculus teacher gave us the above problem, and gave the solution as:
$$y = \tan^{-1}\left(\frac{x\tan z}{10+x}\right) \tag{1}$$
However, I am skeptical. The first few steps of solving this problem were very easy, and I got to where:
$$\tan y = \frac{x\tan z}{10+x} \tag{2}$$
However, it seems to me that you cannot simply jump from there to the answer my teacher gave -- that is, that given the statement in the previous line it need not be true (although it could be true) that $y$ equals what it equals in the teacher's answer above.
This is because of the restricted range of the arctangent/inverse tangent function: $-\frac{\pi}{2}<y<\frac{\pi}{2}$.
Therefore, aren't there an infinite number of answers to the question? (I'm not entirely sure that my "need not be true" statement above is actually the logically correct way of framing my concern, but hopefully you all get a sense of what I'm getting at.)