How do I find the product expansion of $\cos z - \sin z$
We have $\cos z = \sin z$ iff $z = \pi/4 + k \pi$ where $k$ is an integer. The sequence
$\sum (r/(|\pi/4 + k \pi|)^2$ converges
For some sequence of integers $p_n$, $e^{g(z)} \prod_{n \in {\bf Z}} E_{p_n}({z \over \pi/4 + n \pi})) = \cos z - \sin z$ where $E_{p_n}$ is the Weierstrass factor.
and ${p_n} $ = the constant sequence 1 makes the product converge. However unlike the product for sin we don't have cancellation
From Hadamard we have that $g(z)$ is of the form $az + b$ if we use the constant sequence 1 for the p.
$\cos(z) - \sin(z) = - \sqrt{2}\; \sin(z - \pi/4)$, then use the infinite product formula for the sine.