The value of frac{tan x}{tan 3x},whenever def | vedclass

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(A) Let $y = \frac{	an x}{	an 3x}$.Using the formula $	an 3x = \frac{3	an x - 	an^3 x}{1 - 3	an^2 x}$,we get:$y = \frac{	an x}{\frac{3	an x -

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$1/3$ and $3$

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(A) Let $y = \frac{	an x}{	an 3x}$.Using the formula $	an 3x = \frac{3	an x - 	an^3 x}{1 - 3	an^2 x}$,we get:$y = \frac{	an x}{\frac{3	an x - 	an^3 x}{1 - 3	an^2 x}} = \frac{1 - 3	an^2 x}{3 - 	an^2 x}$.Let $t = 	an^2 x$,where $t \ge 0$ and $t 
eq 3$ (since $	an 3x$ must be defined).Then $y = \frac{1 - 3t}{3 - t}$.To find the range,we solve for $t$ in terms of $y$:$y(3 - t) = 1 - 3t \implies 3y - yt = 1 - 3t \implies 3t - yt = 1 - 3y \implies t(3 - y) = 1 - 3y$.$t = \frac{1 - 3y}{3 - y} = \frac{3y - 1}{y - 3}$.Since $t = 	an^2 x \ge 0$,we have $\frac{3y - 1}{y - 3} \ge 0$.Using the wavy curve method,the expression is $\ge 0$ when $y \le 1/3$ or $y \ge 3$.Thus,$y$ never lies between $1/3$ and $3$.

The value of $\frac{\tan x}{\tan 3x}$,whenever defined,never lies between:

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