2000 STEP II, Q4 - Complex Numbers

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…