Proving that $\frac{sinx}{x}=(1-\frac{x^2}{\pi^2})(1-\frac{x^2}{4\pi^2})(...)$

Question

I am faced with explaining to a bunch of people who have taken a course in real analysis, but no course in complex analysis, why $\sin(x)=x\prod_{n\geq1}(1-x^2/\pi^2n^2)$. I vaguely remember being taught a proof of this of the following form: check product converges, and then observe (because sin has simple zeros) that the ratio of the two sides is holomorphic with no zeros or poles; now check that its growth at infinity is not too large (to rule out it being $e^x$ or whatever) and deduce the ratio is a constant, which must be 1". But this "growth" trick uses some complex analysis which they don't know (and in writing this question I realise that I'm no longer sure of it either). Is there any way I can get away with just real analysis? I am happy to use use basic properties of complex numbers and even radius of convergence, but I want to avoid Cauchy's (integral) theorem and beyond.