Answer
$\frac{\sqrt 2+\sqrt 3}{4}$
Work Step by Step
Use the Product to Sum Formula, we have $cos37.5^{\circ}cos7.5^{\circ}=\frac{1}{2}(cos(37.5^{\circ}+7.5^{\circ})+cos(37.5^{\circ}-7.5^{\circ}))=
\frac{1}{2}(cos(45^{\circ})+cos(30^{\circ}))=\frac{1}{2}(\frac{\sqrt 2}{2}+\frac{\sqrt 3}{2})=\frac{\sqrt 2+\sqrt 3}{4}$
