Given $A + B + C + D = 2\pi$, are there special trigonometric identities that concern these four angles? If possible, I would like to know whether the above relation has any implications on any of the three sums below:
$$\sin A + \sin B + \sin C + \sin D,$$ $$\cos A + \cos B + \cos C + \cos D,$$ $$\tan A + \tan B + \tan C + \tan D$$
One (close) example is Ptolemy's theorem: If $w + x + y + z = \pi$, then $\sin(w + x)\sin(y + z) = \sin w \sin y + \sin x \sin z$.
I was doing a bit of work of cyclic quadrilaterals and I thought that any of such identities will be very useful to me.