Solve this equation

Geometry Level 3

Find the number of solutions in ( − π 2 , π 2 ) \large{\left(-\frac{\pi}{2},\frac{\pi}{2} \right)} for the equation

3 tan ⁡ 2 x − 4 tan ⁡ 3 x = tan ⁡ 2 3 x tan ⁡ 2 x \large{3 \tan 2x-4 \tan 3x=\tan^{2} 3x \tan 2x}


The answer is 3.

Tanishq Varshney by Tanishq Varshney , 23, India

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1 solution

Tanishq Varshney
Nov 9, 2015

3 ( tan ⁡ 2 x − tan ⁡ 3 x ) = tan ⁡ 3 x ( 1 + tan ⁡ 3 x tan ⁡ 2 x ) \large{3(\tan 2x-\tan 3x)=\tan 3x(1+\tan 3x \tan 2x)}

− 3 ( tan ⁡ 3 x − tan ⁡ 2 x 1 + tan ⁡ 3 x tan ⁡ 2 x ) = tan ⁡ 3 x \large{-3\left(\frac{\tan 3x-\tan 2x}{1+\tan 3x \tan 2x}\right) =\tan 3x}

− 3 tan ⁡ x = 3 tan ⁡ x − tan ⁡ 3 x 1 − 3 tan ⁡ x \large{-3 \tan x =\frac{3 \tan x-\tan ^{3} x}{1-3 \tan x}}

tan ⁡ x = 0 tan ⁡ 2 ( x ) = 3 5 \large{\tan x=0 \qquad \quad \tan^{2} (x)=\frac{3}{5}}

x = 0 , ± arctan ⁡ ( 3 5 ) \large{x=0, \pm \arctan \left(\sqrt{\frac{3}{5}} \right)}

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