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I have been trying to solve this equation for over a week now: $$\tan5x-2\tan3x=\tan3x\tan5x$$

I found one solution $x=k\pi$ but I cannot prove that this is the only solution. It is equivalent to: $$\sin 5x \cos 3x - 2 \sin 3x \cos 5x = \sin 3x \sin 5x$$ $$\sin 2x =\sqrt 2 \sin 3x \sin(5x+ \frac{\pi}{4})$$ $$2\sin x \cos x =2\sqrt 2 \sin x \cos (2x +1) \sin(5x+ \frac{\pi}{4})$$ So one solution is $x=k\pi$. But how to proceed with the remaining: $$\sqrt 2 \cos x = 2\cos (2x +1) \sin(5x+ \frac{\pi}{4}) $$ It seems impossible to use derivatives to prove that it has no solution because the argument of the third cosine is shifted by $\frac{\pi}{4}$

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    What does tg5x mean ?2012-11-24
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    is tg= trigonometric?2012-11-24
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    Corrected. It means tan5x2012-11-24
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    Wolfram Alpha thinks there are more solutions (none of which --apart from the multiplies of $\pi$-- are very attractive): http://www.wolframalpha.com/input/?i=roots+of+tan%285x%29-2+tan%283x%29-+%28tan%283x%29%29%28tan%285x%29%292012-11-24
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    @Blue: do you mind making your comment as an answer?2013-01-29

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