Solve: $tan 2 x - tan 3 x = 0$

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Solution

$tan 2 x - tan 3 x = 0 \Rightarrow \frac{2 tan x}{1 - {tan}^{2} x} - \frac{3 tan x - {tan}^{3} x}{1 - 3 {tan}^{2} x} = 0 L e t tan x = a \Rightarrow \frac{2 a}{1 - a^{2}} - \frac{3 a - a^{3}}{1 - 3 a^{2}} = 0 \Rightarrow a^{4} + 2 a^{2} + 1 = 0 \Rightarrow {(a^{2} + 1)}^{2} = 0 \Rightarrow a^{2} + 1 = 0 \Rightarrow a^{2} = - 1$

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