Existence of entire function

Question

Show that there exists an entire function $g(z)$ such that $$g(z)=\frac{\sin(5z)-5\sin(z)}{(z-\pi)^3}$$ for all $z$ $\epsilon$ C\ {$\pi$}

My idea is that since sine functions are entire, and $(z-\pi)^3$ is entire on all $z$ $\epsilon$ C\ {$\pi$}, the function $g(z)$ is entire on all $z$ $\epsilon$ C\ {$\pi$}. I know there's something wrong with my proof but I can't identify where, or think of another approach to this question.

Answer 1

Let $f(z):=\frac{\sin(5z)-5\sin(z)}{(z-\pi)^3}$ and show that $\sin(5z)-5\sin(z)$ has at $z=\pi$ a zero of order $3$. Then $f$ has at $z=\pi$ a removable singularity.

Answer 2

May be a way to approach the problem.

If you use $$\sin(5x)=5\sin(x)-2\sin^3(x)+16\sin^5(x)$$ then $$\sin(5x)-5\sin(x)=-2\sin^3(x)+16\sin^5(x)=-2\sin^3(x)(1-8\sin^2(x))$$