Answer
The fault in the solution given in the exercise is that by dividing the equation by $\sin x$, it has lost a possibility, a solution, which is $\sin x=0$.
Work Step by Step
$$\sin^2x-\sin x=0$$
So the next step states that the equation would be divided by $\sin x$ to become
$$\sin x-1=0$$
However, a closer analysis would show that, if we divide the equation by $\sin x$ like this:
$$\frac{\sin^2x}{\sin x}-\frac{\sin x}{\sin x}=\frac{0}{\sin x}$$
This would create a totally different equation, which does not allow for $\sin x=0$. In other words, here in this equation, $\sin x\ne0$.
That means a possibility $\sin x=0$ has been eliminated from the original equation. In fact, if we replace $\sin x=0$ into the original equation, we get
$$0^2-0=0$$
which means $\sin x=0$ is a solution of the equation.
Therefore, the fault in the solution given in the exercise is that by dividing the equation by $\sin x$, it has lost a possibility, a solution, which is $\sin x=0$.
