Tan Summation 1

Geometry Level 5

∑ r = 0 44 tan ⁡ 2 ( 2 r + 1 ) ∘ = ? \large \sum_{r=0}^{44} \tan^2 (2r+1)^\circ = \, ?


The answer is 4005.

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Ashish Gupta by Ashish Gupta , 22, India

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1 solution

Otto Bretscher
Apr 12, 2016

We know that

S ( n ) = ∑ k = 1 n − 1 tan ⁡ 2 ( k π 2 n ) = ( n − 1 ) ( 2 n − 1 ) 3 S(n)=\sum_{k=1}^{n-1}\tan^2\left(\frac{k\pi}{2n}\right)=\frac{(n-1)(2n-1)}{3}

Here the answer is S ( 90 ) − S ( 45 ) = 4005 S(90)-S(45)=\boxed{4005}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Can you please derive that formula I don't. know

roja rani - 4 years, 5 months ago

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