Question

Describe how you would convince a friend that sin(x + y) does not equal sin x + sin y.

Answer 1

If the claim is that $\sin(x+y)=\sin x+\sin y$ for all $x,y$, then all you need to refute it is a single example for which the equality doesn't hold. Take $x=y=\dfrac\pi2$. Then $\sin\left(\dfrac\pi2+\dfrac\pi2\right)=\sin\pi=0$, but $\sin\dfrac\pi2+\sin\dfrac\pi2=1+1=2$.

If the claim is that there is some choice of $x,y$ for which equality holds, then we can recall the angle sum identity for sine:

$\sin(x+y)=\sin x\cos y+\cos x\sin y$

which means that, in order to have equality, we need to choose $x,y$ such that both $\cos y=1$ and $\cos x=1$. We can do that; $\cos x=1$ only when $x$ is an even multiple of $\pi$, i.e. $x\in\{0,\pm2\pi,\pm4\pi,\ldots\}$. Take any two of these and add them together, and you'd end up with yet another even multiple of $\pi$, and so both sides will reduce to 0. Thus equality holds only when $x=2n\pi$ and $y=2m\pi$ for some integers $m,n$.