# How can I prove $\frac{\pi}{4}=\arctan\frac{1}{2}+\arctan\frac{1}{3}$ by using the product $(2+i)(3+i)$?

How can I prove that $$\frac{\pi}{4}=\arctan\frac{1}{2}+\arctan\frac{1}{3}$$ by using the product $$(2+i)(3+i)$$?

What I noticed is that for $$2+i$$ in polar form, $$\arctan\theta_1=\frac{1}{2}$$ and for $$3+i$$ in polar form, $$\arctan\theta_2=\frac{1}{3}$$. However, I don't know how to proceed from here.

• $$(2+i)(3+i) = 5+5i \text{ and } \frac 5 5 = \tan\frac\pi4.$$ Mar 14 '20 at 16:52

hint...use $$\arg(zw)=\arg(z)+\arg(w)$$