Draw $\displaystyle{{315}}^{\circ}\usepackage{cmbright}\Huge\$ in standard position. The terminal side is in the fourth quadrant and the reference angle is $\displaystyle{{45}}^{\circ}\usepackage{cmbright}\Huge\$ . Use the $\displaystyle{{45}}^{\circ}-{{45}}^{\circ}-{{90}}^{\circ}\usepackage{cmbright}\Huge\$ triangle to find the value of this function at the reference angle:

$\displaystyle{{\cos{{45}}}}^{\circ}=\frac{{1}}{\sqrt{{2}}}=\frac{{1}}{\sqrt{{2}}}\cdot\frac{\sqrt{{2}}}{\sqrt{{2}}}=\frac{\sqrt{{2}}}{{2}}\usepackage{cmbright}\Huge\$

Use the signs of x and y in the fourth quadrant to determine the sign of cosine:

$\displaystyle{{\cos{{315}}}}^{\circ}=\frac{{x}}{{1}}=\frac{{{\left(+\right)}}}{{{\left(+\right)}}}={\left(+\right)}\usepackage{cmbright}\Huge\$

So: $\displaystyle{{\cos{{315}}}}^{\circ}=\frac{\sqrt{{2}}}{{2}}\usepackage{cmbright}\Huge\$