Question

If $A, B \in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, then the maximum value of $(\cos A-\cos B)^{2}+(\sin A-\sin B)^{2}$ is_______.

Answer 1

$(\cos A-\cos B)^{2}+(\sin A-\sin B)^{2}$
$=\cos ^{2} A+\cos ^{2} B-2 \cos A \cos B+\sin ^{2} A+\sin ^{2} B-2 \sin A \sin B$
$=2-2 \cos (A-B)$
$=4 \sin ^{2}\left(\frac{A-B}{2}\right)$
Also, $A, B \in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$
$\Rightarrow-\frac{\pi}{2} \leq A-B \leq \frac{\pi}{2}$
$\Rightarrow-\frac{\pi}{4} \leq \frac{A-B}{2} \leq \frac{\pi}{4}$
$\Rightarrow$ maximum value is $4 \times\left(\frac{1}{\sqrt{2}}\right)^{2}=2$