Solve : $tan 2 x - tan 3 x$

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Solution

Given $tan 2 x - tan 3 x$
$\to tan 2 x - tan 3 x = tan (- x) (1 + tan 2 x tan 3 x) = 0$
So, either $tan x = 0 \Rightarrow x = n π \forall n \in z$
OR
$\to tan 2 x tan 3 x = - 1$
$\to sin 2 x sin 3 x + cos 2 x cos 3 x = 0$
$\to cos x = 0$
$\to x = (2 n + 1) \frac{π}{2} \forall n \in z$

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