Prove that $tan 3 x tan 2 x tan x = tan 3 x - tan 2 x - tan x$ .

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Solution

We know that
$tan 3 x = tan (2 x + x) = \frac{tan 2 x + tan x}{1 - tan 2 x tan x}$
$tan 3 x - tan 3 x tan 2 x tan x = tan 2 x + tan x$
$tan 3 x tan 2 x tan x = tan 3 x - tan 2 x - tan x$

Hence, proved.

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