I was reading about mathematical coincidences recently, and I came across the borderline unbelievable fact that $$\int_{0}^{\infty} \cos(2x)\prod_{n=1}^{\infty} \cos\Big(\frac{x}{n}\Big) dx\approx \frac{\pi}{8}$$ to $42$ decimal places!
But I'm familiar with the concept of numerical "coincidences" that nevertheless have some sort of actually cognizable reason behind them, like the fact that $e^{\pi\sqrt{163}}\approx 262537412640768743.999999999999\approx 262537412640768744$ due to something involving Heegner numbers I don't fully understand, so I know this could possibly be more than the above example- it even has a far greater degree of accuracy, by a factor of about $10^{11}$ if my back-of-the-envelope calculation is right. So my question is: is the above a fact that seems coincidental but actually has a very good rationale, or is it a pure coincidence like the fact that $e^{\pi}-\pi\approx 19.999099979\approx 20$?