How to prove that $\tan1^{°}+\tan7^{°}+\dots+\tan175^{°}=-30\sqrt3$?

Asked Aug 07 '14 at 09:24

Active Dec 27 '17 at 05:20

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I noticed that $1^{°},7^{°},\dots,175^{°}$ form an arithmetic.

edited Dec 27 '17 at 05:20

asked Aug 07 '14 at 09:24

piteer

Can you post the third term in the sequence? Do you have it with you? – MonK Aug 07 '14 at 09:33
Nevermind, got it. Assuming the angles are in an AP. – MonK Aug 07 '14 at 09:38
@piteer, Set $\theta=1^\circ,n=30$ to find the sum to be $$-30\cot\left(30\cdot90^\circ+30\cdot1^\circ\right)$$
Now, $$\cot\left(30\cdot90^\circ+30\cdot1^\circ\right)=\cot\left(15\cdot180^\circ+30^\circ\right)=\cot30^\circ$$
– lab bhattacharjee Aug 07 '14 at 16:20
@piteer, The previous comment actually missed the link : http://math.stackexchange.com/questions/346368/sum-of-tangent-functions-where-arguments-are-in-specific-arithmetic-series – lab bhattacharjee Aug 08 '14 at 15:54

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