Answer
$ \frac{\sqrt {2-\sqrt 3}}{2}$
Work Step by Step
$sin165^\circ=sin15^\circ=sin\frac{30^\circ}{2}=\sqrt {\frac{1-cos30^\circ}{2}}
=\sqrt {\frac{1-\sqrt 3/2}{2}}=\frac{\sqrt {2-\sqrt 3}}{2}$
Note: this answer is the same as $\frac{\sqrt 6-\sqrt 2}{4}$
