Animated Solution for Mathematics - Trigonometry: Show that the value of $\frac{\tan x}{\tan 3x}$, wherever defined never lies between $1/3$ and 3.

The problem asks us to prove that the expression $y=\frac{\tan x}{\tan 3x}$ never takes a value in the interval $(\frac{1}{3},3)$. To begin, we utilize the standard triple angle identity for the tangent function:

Substituting this identity into our expression for $y$, we obtain:

Assuming $\tan x eq 0$, we can cancel $\tan x$ from the numerator and the denominator. This simplification yields a much more manageable rational function:

To further simplify the algebra, we introduce the substitution $t={\tan }^{2}x$. This transforms the expression into:

We must respect the domain of our substitution. Since $t$ is the square of a real number $\tan x$, it is strictly required that $t\ge 0$. This constraint is vital for determining the valid range of $y$.

To find the range, we invert the function to solve for $t$ in terms of $y$:

Factoring out $t$, we arrive at:

Given our constraint $t\ge 0$, we must satisfy the following inequality:

Using the wavy curve method, we identify the critical points at $y=\frac{1}{3}$ and $y=3$. Testing the intervals, we find that the expression is non-negative when $y\le \frac{1}{3}$ or $y>3$.

Consequently, the range of $y$ is $(-\infty ,\frac{1}{3}]\cup (3,\infty )$. This confirms that the expression $y$ never takes a value in the interval $(\frac{1}{3},3)$.