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

Answer: 2

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