Three Embedded Inequalities

Let $A, B, C$ be the interior angles of a triangle and $x, y, z$ real numbers. For each of the following, determine (with proof or counterexample) whether it holds for all such $A, B, C, x, y, z$:

1. $x^2 + y^2 + z^2 - 2yz\sin A - 2zx\sin B + 2xy\cos C \geqslant 0$; 2. $x^2 + y^2 + z^2 - 2yz\sin A + 2zx\sin B - 2xy\cos C \geqslant 0$; 3. $x^2 + y^2 + z^2 + 2yz\cos A + 2zx\cos B - 2xy\cos C \geqslant 0$.