$3 tan 2 x - 4 tan 3 x = {tan}^{2} 3 x tan 2 x$ .

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Solution

We write the given equation as:
$3 tan 2 x - 3 tan 3 x - tan 3 x + {tan}^{2} 3 x tan 2 x$
or $\frac{3 (tan 2 x - tan 3 x)}{1 + tan 2 x tan 3 x} = tan 3 x$
or $3 tan (2 x - 3 x) = tan 3 x$
or $3 tan x + tan 3 x = 0$
If we put $tan x = t$ , then from above we get
$3 t + \frac{3 t - t^{3}}{1 - 3 t^{2}} = 0$ or $6 t - 10 t^{3} = 0$
$∴ t = 0$ or $t = \pm \sqrt{(3 / 5)}$
where $t = tan x$ $∴ x = n π$ or $n π \pm α$
where $α = {tan}^{- 1} \sqrt{(3 / 5)}$ .

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