I have found a very weird looking infinite product:
$$2+\sqrt{3}-\sqrt{2}-\sqrt{6}=\dfrac{-1\cdot1\cdot23\cdot25\cdot23\cdot25\cdot47\cdot49\cdot47\cdot49\cdot71\cdot73\cdot71\cdot73\cdots}{2\cdot4\cdot20\cdot22\cdot26\cdot28\cdot44\cdot46\cdot50\cdot52\cdot68\cdot70\cdot74\cdot76\cdots}$$
The numerator terms(after the first two) follow the sequence, $(+22,+2,-2,+2)*n$ and the denominators(after the first two) follow $(+16,+2,+4,+2)*n$
I considered:
$$\cos(x)+\sin(x)-\dfrac{\sqrt{3}+1}{2}=\dfrac{1-\sqrt{3}}{2}\cdot\prod_{i=1}^{\infty}(1-\dfrac{x}{Root_i})$$ $$\cos(x)+\sin(x)-\dfrac{\sqrt{3}+1}{2}=\dfrac{1-\sqrt{3}}{2}\cdot(1-\dfrac{6x}{\pi})(1-\dfrac{6x}{2\pi})(1+\dfrac{6x}{10\pi})(1+\dfrac{6x}{11\pi})(1-\dfrac{6x}{13\pi})(1-\dfrac{6x}{14\pi})\cdots$$ And substituted the value $x=\dfrac{\pi}{4}$
The roots of the equation being $\dfrac{\pi}{3}+2n\pi,\dfrac{\pi}{6}+2n\pi$
The value after a few products on the RHS equals $-0.1304895$. The value of the LHS is approximately $-0.13165249758$
However I'm not sure if the Weierstrass factorization can be manipulated in this way. Please verify this result and rectify any errors.