Answer
$1$
Work Step by Step
Given expression is-
$\sin^{2} 45^{\circ} + \cos^{2} 45^{\circ}$
= $(\sin 45^{\circ})^{2} + (\cos 45^{\circ})^{2}$
= $(\frac{1}{\sqrt 2})^{2} + (\frac{1}{\sqrt 2})^{2} $
(Substituting exact values)
= = $(\frac{1}{2}) + (\frac{1}{2}) $
= $1$