Answer
$\sqrt {\frac{2-\sqrt 2}{2}}$
Work Step by Step
Use the Sum to Product Formula, we have
$cos67.5^{\circ}+cos22.5^{\circ}=2cos\frac{67.5^{\circ}+22.5^{\circ}}{2}cos\frac{67.5^{\circ}-22.5^{\circ}}{2}
=2cos\frac{90^{\circ}}{2}cos\frac{45^{\circ}}{2}=\sqrt 2cos\frac{45^{\circ}}{2}$
Use the Half-Angle Fomula, we get $cos\frac{45^{\circ}}{2}=\sqrt {\frac{1-cos45^{\circ}}{2}}
=\sqrt {\frac{1-\frac{\sqrt 2}{2}}{2}}=\sqrt {\frac{2-\sqrt 2}{4}}$
Thus $cos67.5^{\circ}+cos22.5^{\circ}=\sqrt 2\sqrt {\frac{2-\sqrt 2}{4}}=\sqrt {\frac{2-\sqrt 2}{2}}$
