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Question Number 164720 by mathlove last updated on 21/Jan/22

$$tan3x=5tanx$$$$x=?$$

Answered by leonhard77 last updated on 21/Jan/22

$$\tan 3x=\frac{3\tan x-\tan {}^{3}x}{1-3\tan {}^{2}x}=5\tan x$$$$\Rightarrow \tan x(\frac{3-\tan {}^{2}x}{1-3\tan {}^{2}x}-5)=0$$$$\Rightarrow \tan x=0,x=n\pi$$$$\Rightarrow \frac{3-\tan {}^{2}x-5+15\tan {}^{2}x}{1-3\tan {}^{2}x}=0$$$$\Rightarrow 14\tan {}^{2}x=2$$$$\Rightarrow \tan {}^{2}x=\frac{1}{7}$$

Answered by mr W last updated on 21/Jan/22

$$\frac{\tan 2x+\tan x}{1-\tan 2x\tan x}=5\tan x$$$$\frac{\frac{2\tan x}{1-{\tan }^{2}x}+\tan x}{1-\frac{2\tan x}{1-{\tan }^{2}x}\tan x}=5\tan x$$$$\Rightarrow \tan x=0\Rightarrow x=k\pi$$$$\frac{\frac{2}{1-{\tan }^{2}x}+1}{1-\frac{2{\tan }^{2}x}{1-{\tan }^{2}x}}=5$$$$\frac{3-{\tan }^{2}x}{1-3{\tan }^{2}x}=5$$$${\tan }^{2}x=\frac{1}{7}$$$$\tan x=\pm \frac{\sqrt{7}}{7}$$$$\Rightarrow x=k\pi \pm {\tan }^{-1}\frac{\sqrt{7}}{7}$$

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