The period of $\displaystyle{y}={a}{\cos{{b}}}{x}\usepackage{cmbright}\Huge\$ is given by $\displaystyle\frac{{{2}\pi}}{{\left|{b}\right|}}\usepackage{cmbright}\Huge\$ . So the period of $\displaystyle{y}={\cos{{4}}}{x}\usepackage{cmbright}\Huge\$ is:

$\displaystyle\frac{{{2}\pi}}{{\left|{4}\right|}}=\frac{{{2}\pi}}{{4}}=\frac{\pi}{{2}}\usepackage{cmbright}\Huge\$