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August 27th, 2016, 04:14 PM  #1 
Newbie Joined: Aug 2016 From: Brazil Posts: 8 Thanks: 0  indicate first three positive solutions
Solve $\displaystyle \tan(3x) + \cot(x) = \tan(x) + \cot(3x)$. Indicate first three positive solutions. Answer: $\displaystyle \frac{\pi }{4},\frac{3 \pi}{4},\frac{5 \pi}{4} $ Last edited by skipjack; August 28th, 2016 at 02:01 AM. 
August 28th, 2016, 12:47 AM  #2 
Newbie Joined: Dec 2006 Posts: 5 Thanks: 2 
$\displaystyle \tan 3x\tan x=\cot 3x\cot x$ $\displaystyle \frac{\sin 2x}{\cos x\cos 3x}=\frac{\sin 2x}{\sin x\sin 3x}$ $\displaystyle \sin 2x(\cos x\cos 3x+\sin x\sin 3x)=\Rightarrow \sin 2x\cos 2x=0$ Multiply both members by 2 $\displaystyle \sin 4x=0\Rightarrow x=\frac{k\pi}{4}$ The first three positive solutions are obtained for $\displaystyle k=1, \ 3, \ 5$ 

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