Bijective Reckoning

Apr 15, 2020
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mathematics

This lovely little STEP question looks at how De Moivres theorem can be used to show some interesting identities

Today, I’d like to look a little bit at this lovely problem from the 2000 STEP II paper:

Problem.Prove that $$ (\cos \theta+\mathrm{i} \sin \theta)(\cos \phi+\mathrm{i} \sin \phi)=\cos (\theta+\phi)+\mathrm{i} \sin (\theta+\phi) $$ and that, for every positive integer $n$ $$ (\cos \theta+\mathrm{i} \sin \theta)^{n}=\cos n \theta+\mathrm{i} \sin n \theta $$ By considering $(5-i)^{2}(1+\mathrm{i})$, or otherwise, prove that $$ \arctan \left(\frac{7}{17}\right)+2 \arctan \left(\frac{1}{5}\right)=\frac{\pi}{4} $$ Prove also that $$ 3 \arctan \left(\frac{1}{4}\right)+\arctan \left(\frac{1}{20}\right)+\arctan \left(\frac{1}{1985}\right)=\frac{\pi}{4} $$

To dissect it a little bit, it begins with a standard proof of De Moivre’s theorem for positive integer exponents, and then we proceed to show things about angles, by relating complex numbers and their arguments. Interesting indeed…