Answer
$\frac{\sqrt{2}-1}{4}$
Work Step by Step
Use the second Product-to-Sum Formula, $\cos u\sin v=\frac{1}{2}[\sin(u+v)-\sin(u-v)]$.
$\cos37.5^\circ\sin7.5^\circ$
$=\frac{1}{2}[\sin(37.5^\circ+7.5^\circ)-\sin(37.5^\circ-7.5^\circ)]$
$=\frac{1}{2}(\sin 45^\circ-\sin 30^\circ)$
$=\frac{1}{2}(\frac{\sqrt{2}}{2}-\frac{1}{2})$
$=\frac{1}{2}(\frac{\sqrt{2}-1}{2})$
$=\frac{\sqrt{2}-1}{4}$
