Sum Of Tangent Squared?

Geometry Level 4

$\large \sum_{r=0}^{89}\tan^2(r^\circ)$

If the value of the above summation can be expressed as $\dfrac{p}{q}$ , where $p$ and $q$ are coprime positive integers, find $p+q$

Clarification : Angles are measured in degrees.

This problem is part of my set: Geometry

1 solution

Otto Bretscher
Mar 18, 2016

Use the formula $\sum_{k=0}^{n-1}\tan^2\left(\frac{k\pi}{2n}\right)=\frac{(n-1)(2n-1)}{3}$ with $n=90$ . Find various proofs in the solutions here

Tu comprends la langue français?

Department 8 - 5 years, 2 months ago

On parle français en Suisse (mais ma langue maternelle est suisse-allemand).

Otto Bretscher - 5 years, 2 months ago

Didn't know that such a problem had been already given once.

A Former Brilliant Member - 5 years, 2 months ago

It's good to remind people of this lovely formula (and its proof) from time to time ;)

Otto Bretscher - 5 years, 2 months ago

By the way, just asking, how is it that your question is level 4 while mine (pretty much same as yours) is level 5?

A Former Brilliant Member - 5 years, 2 months ago

@A Former Brilliant Member – These "Levels" seem pretty random to me anyway

Otto Bretscher - 5 years, 2 months ago