Which of the following is not a possible value of $f(x)=\tan 3x\cot x$?
Step-by-Step Solution
Key Concept: Express the composite trigonometric function in terms of a single variable and find its range using inequality constraints.
Given $f(x) = \tan 3x \cdot \cot x = \frac{\sin 3x}{\cos 3x} \cdot \frac{\cos x}{\sin x} = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x} \cdot \frac{1}{\tan x} = \frac{3 - \tan^2 x}{1 - 3\tan^2 x}$. Let $y = \frac{3-\tan^2 x}{1-3\tan^2 x}$, then $\tan^2 x = \frac{y-3}{3y-1}$. Since $\tan^2 x \geq 0$, we require $\frac{y-3}{3y-1} \geq 0$, giving $y \in (-\infty, -\frac{1}{3}) \cup (3, \infty)$.
Correct Answer: 1,2