Saturday, 22 March 2014

reference request - Proofs without words

$$arctan frac{1}{3} + arctan frac{1}{2} = arctan 1$$



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It's easy to generalize this to



$$ arctan frac{1}{n} + arctan frac{n-1}{n+1} = arctan 1, text{ for } n in mathbb{N}$$



which can further be generalized to



$$ arctan frac{a}{b} + arctan frac{b-a}{b+a} = arctan 1, text{ for } a,b in mathbb{N}, a leq b $$



Edit: A similar result relating Fibonacci numbers to arctangents can be found here and here.

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