Matematicamente.it

Sono tre famiglie di rette, in quanto: \[
\begin{aligned}
& \sin(x+y) = \sin(x)+\sin(y) \\
\\
& 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x+y}{2}\right) = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) \\
\\
& 2\sin\left(\frac{x+y}{2}\right)\left[\cos\left(\frac{x+y}{2}\right)-\cos\left(\frac{x-y}{2}\right)\right] = 0 \\
\\
& -4\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x}{2}\right)\sin\left(\frac{y}{2}\right) = 0 \\
\\
& \boxed{\frac{x+y}{2}=k\pi \, \vee \, \frac{x}{2}=k\pi \, \vee \, \frac{y}{2}=k\pi, \; k \in \mathbb{Z}\,} \\
\end{aligned}
\]