Find $\tan x$

If $$a \cos x + b \sin x = c$$

Find the possible values of $\tan x$ in terms of $a, b, c$.

No calculus IVT for quadratic.

Suppose $$f(x) = x^2 + bx + c$$ and that $f(0) \lt 0$ and $f(1) \gt 0$.

Since $f$ is continuous, by the intermediate value theorem there is some $c \in (0,1)$ such that $f(c) = 0$.

The question here is to prove that in an elementary way, without using any calculus concepts.