\cos 165\degree =\cos(120\degree +45\degree )

=\cos 120\degree \cos45\degree -\sin 120\degree \sin 45\degree

=-\dfrac{\sqrt{3} }{2} \cdotp \dfrac{\sqrt{2} }{2} -\dfrac{\sqrt{3} }{2} \cdotp \dfrac{\sqrt{2} }{2}

=-\dfrac{\sqrt{6} }{4}