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Finite sum formula for tangent (trigonometry) 
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#1
Mar2013, 01:28 AM

P: 84

Hi everyone, I've been looking for the finite sum formulae of trig functions. I've found the easiest ones (sine and cosine). But the one for the tangent seems to be very hard. No mathematical tricks work. Plus I've looked it up on the internet. Nothing. I will greatly appreciate your help. Thanks in advance.
tan x + tan (2x) + tan (3x) + ... + tan (nx) = ??? 


#2
Mar2013, 03:10 PM

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#3
Mar2013, 03:42 PM

P: 84




#4
Mar2013, 08:07 PM

P: 40

Finite sum formula for tangent (trigonometry)
I found in a textbook that [itex]tan(x)[/itex] can be written as an indefinite sum:
[itex]\sum_x \tan ax = i x\frac1a \psi _{e^{2 i a}}\left(x\frac{\pi }{2 a}\right) + C \,,\,\,a\ne \frac{n\pi}2[/itex] where [itex]\psi_q(x)[/itex] is the qdigamma function. Computing "sum k from 1 to n of tan(k*x)" in WolframAlpha results into something much more complicated, but an answer is given. 


#5
Mar2013, 08:20 PM

P: 84




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