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Is there any method to find value of $$\tan(1^{\circ})+\tan(2^{\circ})+\tan(3^{\circ})+\dots+\tan(89^{\circ})$$ without using calculator.

To find the same sum for sine and cosine , I used De-Movier's Theorem but here I can't see any approach working. Any suggestions?

Mathematics
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  • Perhaps the tangent addition formula might help? It might take some time to fully calculate everything though. – Horus Mar 03 '16 at 17:54
  • @tatan I don't think so. $tan(45)$ alone is 1. In case of product it will be 1 – Mathematics Mar 03 '16 at 17:56
  • This is related to that question, but not the same. – Robert Israel Mar 03 '16 at 18:02
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    The answer here is not particularly nice, I think: it's the root near $265.017$ of a polynomial of degree $24$, irreducible over the rationals (and too big to fit in a comment). – Robert Israel Mar 03 '16 at 18:37
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    An approximation can be found by considering the sum as a Riemann sum for the integral $\frac{360}{2\pi}\int_0^{\frac{2\pi}{360}89}\tan(x){\rm d}x = -\frac{180 \log \left(\sin \left(\frac{\pi }{180}\right)\right)}{\pi } \approx 231.95$. – Winther Mar 04 '16 at 02:46
  • Hint: Find the coefficient of the $x^{88}$ term in the polynomial $$P(x) = (x-\tan 1^{\circ})(x-\tan 2^{\circ})(x-\tan 3^{\circ})\dots(x-\tan 89^{\circ})$$ – John Joy Mar 04 '16 at 13:44

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