Chapter 5 - Section 5.1 - Proving Identities - 5.1 Problem Set - Page 280: 83

$\sin(A+B)=\sin A+\sin B$ is not an identity.

$\sin(30^{o}+60^{o})=\sin 90^{o}=1$ $\displaystyle \sin(30^{o})+\sin(60^{o})=\frac{1}{2}+\frac{\sqrt{3}}{2}=\frac{1+\sqrt{3}}{2}\neq 1$ So, $\sin(A+B)=\sin A+\sin B$ is not an identity. ($A$ counterexample is $A=30^{o}, B=60^{o})$