$$arg(zw)\equiv arg(z)+arg(w) (\text{mod } 2\pi)$$
is a standard property of complex numbers. I know the proof using the polar form or the exponential form.I am attempting to prove it using the $tan^{-1}$ definition of $arg$.
Let there be two complex numbers such that $z=a+bi,w=c+di$. Then, the left hand side equals $$arg(zw)=arg[(a+bi)(c+di)]=arg[ac-bd+(bc+ad)i)]=tan^{-1} \left[ \frac{bc+ad}{ac-db} \right]$$
Similarly, the right hand side equals: $$arg(z)+arg(w)=arg(a+bi)+arg(c+di)=tan^{-1}\left(\frac{b}{a}\right)+tan^{-1}\left(\frac{d}{c}\right)$$ Use the property $tan^{-1}(x)+tan^{-1}(y)=tan^{-1}(\frac{x+y}{1-xy}), xy<1$ $$=tan^{-1}\left(\frac{\frac{b}{a}+\frac{d}{c}}{1-\frac{b}{a}\times\frac{d}{c}}\right)=tan^{-1} \left[ \frac{bc+ad}{ac-db}\right]$$
This is under the assumption:
- $tan^{-1}$ will give the correct argument only for complex numbers in the first and fourth quadrant. Hence, assume these conditions are met.
- We need the product of the angles to be less than one. Else the property changes to:
$$tan^{-1}(x)+tan^{-1}(y)=\pi + tan^{-1}(\frac{x+y}{1-xy}), xy>1$$
Is the above proof correct? Can it be generalized to remove the assumptions?
I appreciate comments on stating the property correctly. However, my question is more on generalizing the proof.