Graphs of $x\sin x=\sin x$ and $x=1$

Question

I put $x\sin x = \sin x$ in desmos and it shows this.

As it seems, this is not the graph of $x=1$.

My question is : Why $\sin x$ can't be cancelled from both sides of $x\sin x =\sin x$ to get $x=1$ ? The graph shows that they are not same.

Answer 1

$x \sin x = \sin x$

There are two cases to consider.

If $\sin x \ne 0$, then you can divide both sides by $\sin x$ and you get $x=1$.

If $\sin x=0$, then you get $ x = n\pi$ where $n$ can have any integer value.

So the solution set is $\{0, 1, \ \pm \pi, \ \pm 2\pi, \ \pm 3\pi, \dots \}$

Answer 2

Because $\sin x$ could vanish. Your equation is equivalent to $$(\sin x)(x-1)=0,$$ which tells us that $x=1$ or $\sin x=0.$ The last equation is a warning signal that things could go wrong if we attempted indiscriminate division by $\sin x.$

PS. In general, one should be careful with taking quotients (in particular, one should bear in mind that the divisor could vanish; if the divisor is identically zero, one cannot divide).