[proofplan]
The identity is a direct computation. We expand the product $zw$ using the distributivity of multiplication in $\mathbb{C}$ over addition and the relation $i^2 = -1$, which separates the result into real and imaginary parts each expressed as a bilinear combination of $\{\cos\theta, \sin\theta, \cos\phi, \sin\phi\}$. We then recognise these bilinear combinations as the right-hand sides of the cosine and sine angle-addition identities, which collapses them into $\cos(\theta+\phi)$ and $\sin(\theta+\phi)$. The modulus and argument statements follow by reading off the resulting polar form.
[/proofplan]
[step:Expand the product using distributivity in $\mathbb{C}$ and $i^2 = -1$]
Writing $z = r\cos\theta + i\,r\sin\theta$ and $w = s\cos\phi + i\,s\sin\phi$, and applying distributivity of multiplication over addition in the field $\mathbb{C}$ together with the defining relation $i^2 = -1$:
\begin{align*}
zw &= \bigl(r\cos\theta + i\,r\sin\theta\bigr)\bigl(s\cos\phi + i\,s\sin\phi\bigr) \\
&= rs\cos\theta\cos\phi + i\,rs\cos\theta\sin\phi + i\,rs\sin\theta\cos\phi + i^2\,rs\sin\theta\sin\phi \\
&= rs\bigl(\cos\theta\cos\phi - \sin\theta\sin\phi\bigr) + i\,rs\bigl(\sin\theta\cos\phi + \cos\theta\sin\phi\bigr).
\end{align*}
The real part is $rs(\cos\theta\cos\phi - \sin\theta\sin\phi)$ and the imaginary part is $rs(\sin\theta\cos\phi + \cos\theta\sin\phi)$.
[guided]
We compute $zw$ from the definitions. Since $\mathbb{C}$ is a field, multiplication is distributive over addition, so we can expand the product of the two binomials $r\cos\theta + i\,r\sin\theta$ and $s\cos\phi + i\,s\sin\phi$ term by term — this is FOIL in $\mathbb{C}$. The only non-real ingredient is $i$, and the only rule we need about it is $i^2 = -1$, which is built into the definition of $\mathbb{C}$.
Expanding:
\begin{align*}
zw &= \bigl(r\cos\theta + i\,r\sin\theta\bigr)\bigl(s\cos\phi + i\,s\sin\phi\bigr) \\
&= (r\cos\theta)(s\cos\phi) + (r\cos\theta)(i\,s\sin\phi) + (i\,r\sin\theta)(s\cos\phi) + (i\,r\sin\theta)(i\,s\sin\phi).
\end{align*}
The four products give, using commutativity of the real scalars $r, s$ and the trigonometric values:
\begin{align*}
zw &= rs\cos\theta\cos\phi + i\,rs\cos\theta\sin\phi + i\,rs\sin\theta\cos\phi + i^2\,rs\sin\theta\sin\phi.
\end{align*}
The fourth term contains $i^2 = -1$, turning it into $-rs\sin\theta\sin\phi$, which is real. Grouping real and imaginary parts:
\begin{align*}
zw &= \underbrace{rs\bigl(\cos\theta\cos\phi - \sin\theta\sin\phi\bigr)}_{\operatorname{Re}(zw)} + i\,\underbrace{rs\bigl(\sin\theta\cos\phi + \cos\theta\sin\phi\bigr)}_{\operatorname{Im}(zw)}.
\end{align*}
At this stage the product is written as a single [complex number](/page/Complex%20Number) in rectangular form $A + iB$ with $A, B \in \mathbb{R}$, where $A$ and $B$ are each a bilinear expression in the sines and cosines of $\theta$ and $\phi$. The next step is to recognise these bilinear expressions.
[/guided]
[/step]
[step:Collapse via the cosine and sine angle-addition identities]
The angle-addition identities state that for all $\theta, \phi \in \mathbb{R}$,
\begin{align*}
\cos(\theta + \phi) &= \cos\theta\cos\phi - \sin\theta\sin\phi, \\
\sin(\theta + \phi) &= \sin\theta\cos\phi + \cos\theta\sin\phi.
\end{align*}
Substituting these into the expression from the previous step:
\begin{align*}
zw &= rs\cos(\theta + \phi) + i\,rs\sin(\theta + \phi) \\
&= rs\bigl(\cos(\theta + \phi) + i\sin(\theta + \phi)\bigr).
\end{align*}
Since $r = |z| > 0$ and $s = |w| > 0$, the right-hand side is the polar form of $zw$ with modulus $rs$ and argument $\theta + \phi$. Reading off the modulus and argument:
\begin{align*}
|zw| &= rs = |z||w|, \\
\arg(zw) &\equiv \theta + \phi = \arg(z) + \arg(w) \pmod{2\pi}.
\end{align*}
The congruence modulo $2\pi$ reflects the standard convention that $\arg$ is defined up to an integer multiple of $2\pi$. This completes the proof.
[guided]
We now translate the bilinear trigonometric expressions obtained above into single trigonometric values at the sum angle $\theta + \phi$. The cosine angle-addition identity says
\begin{align*}
\cos(\theta + \phi) = \cos\theta\cos\phi - \sin\theta\sin\phi
\end{align*}
for all $\theta, \phi \in \mathbb{R}$. This is precisely the bilinear expression that appeared as the real part of $zw$. Similarly, the sine angle-addition identity gives
\begin{align*}
\sin(\theta + \phi) = \sin\theta\cos\phi + \cos\theta\sin\phi,
\end{align*}
which matches the imaginary part. Both identities hold unconditionally on $\mathbb{R} \times \mathbb{R}$, so there is no hypothesis to verify beyond $\theta, \phi \in \mathbb{R}$, which is part of the polar-form assumption on $z$ and $w$.
Substituting both identities into the rectangular form from the previous step:
\begin{align*}
zw &= rs\bigl(\cos\theta\cos\phi - \sin\theta\sin\phi\bigr) + i\,rs\bigl(\sin\theta\cos\phi + \cos\theta\sin\phi\bigr) \\
&= rs\cos(\theta + \phi) + i\,rs\sin(\theta + \phi) \\
&= rs\bigl(\cos(\theta + \phi) + i\sin(\theta + \phi)\bigr).
\end{align*}
This final line is a complex number in polar form: a positive real scalar $rs$ multiplying $\cos\psi + i\sin\psi$ with $\psi := \theta + \phi$. Because $r = |z|$ and $s = |w|$ are strictly positive (the hypothesis $z, w \neq 0$ ensures $r, s > 0$), the scalar $rs$ is strictly positive, so it is a legitimate modulus and the expression is genuinely in polar form — not merely an algebraic rearrangement.
We now extract the two statements. First, the modulus:
\begin{align*}
|zw| = rs = |z||w|,
\end{align*}
since the modulus of a complex number written as $\rho(\cos\psi + i\sin\psi)$ with $\rho > 0$ equals $\rho$. Second, the argument:
\begin{align*}
\arg(zw) \equiv \theta + \phi = \arg(z) + \arg(w) \pmod{2\pi}.
\end{align*}
We write $\equiv \pmod{2\pi}$ because $\arg$ is a multivalued function: the argument of a nonzero complex number is defined only up to integer multiples of $2\pi$. The sum $\theta + \phi$ is one valid argument of $zw$; any other valid argument differs by $2\pi k$ for some $k \in \mathbb{Z}$. This is why the statement of the theorem says "modulo $2\pi$" rather than equality.
This establishes both claims in the theorem and completes the proof.
[/guided]
[/step]
