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August 27th, 2016, 04:14 PM   #1
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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.
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August 28th, 2016, 12:47 AM   #2
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$\displaystyle \tan 3x-\tan x=\cot 3x-\cot x$
$\displaystyle \frac{\sin 2x}{\cos x\cos 3x}=-\frac{\sin 2x}{\sin x\sin 3x}$
$\displaystyle \sin 2x(\cos x\cos 3x+\sin x\sin 3x)=\Rightarrow \sin 2x\cos 2x=0$
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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positive, solutions

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