What is the number of solutions of $(\sin \theta - \cos \theta)^2 = 2$ where $-\pi < \theta < \pi$?

$ABC$ is a triangle such that angle $C = 60^{\circ}$, then what is $\frac{\cos A + \cos B}{\cos \left(\frac{A - B}{2}\right)}$ equal to?

What is $\sqrt{15 + \cot^2 \left( \frac \pi 4 - 2 \cot^{-1} 3 \right)}$ equal to?

What is the value of $\sin10^{\circ} \cdot \sin50^{\circ} + \sin50^{\circ} \cdot \sin250^{\circ} + \sin250^{\circ} \cdot \sin10^{\circ}$ equal to?