I have derived the following series for $t \in \mathbb{R} \setminus \mathbb{Z}$ $$ \pi \cot(\pi t) = \frac{1}{t} + \sum_{k=1}^\infty \frac{2z}{z^2-k^2}, $$ and from that through uniform convergence and integration proved $$ \frac{\sin(\pi t)}{\pi t} = \prod_{k=1}^\infty \Big( 1-\frac{t^2}{k^2} \Big), $$ again for $t \in \mathbb{R} \setminus \mathbb{Z}$.
The question is how could I now extend the Euler's product formula to the whole $\mathbb{R},$ so also to the integers, rigorously? Any help is very much appreciated.
The LHS can clearly be extended to a continuous function of $t$ defined on all of $\Bbb{R}$ (at $t=0$, define it to be $1$). The RHS is continuous since the product converges normally on compact subsets of $\Bbb{R}$. Continuous functions which agree on a dense set agree everywhere, thus both sides are equal on all of $\Bbb{R}$.
In fact, from here, if you replace $t\in\Bbb{R}$ by $z\in\Bbb{C}$, then both sides actually define entire functions, and of course, entire functions which agree on $\Bbb{R}$ agree on all of $\Bbb{C}$, so the same formula holds on $\Bbb{C}$ as well.