Determine the value of \[ \cos 23^{\circ} \cos 68^{\circ}+\sin 23^{\circ} \sin 68^{\circ} \]

We begin with the expression \[ \cos 23^{\circ} \cos 68^{\circ} + \sin 23^{\circ} \sin 68^{\circ}. \] This can be simplified using the cosine of a sum identity: \[ \cos(a - b) = \cos a \cos b + \sin a \sin b. \] In our case, let \( a = 68^{\circ} \) and \( b = 23^{\circ} \). Hence, the expression becomes: \[ \cos(68^{\circ} - 23^{\circ}). \] Now we calculate \( 68^{\circ} - 23^{\circ} \): \[ 68^{\circ} - 23^{\circ} = 45^{\circ}. \] Thus, we have \[ \cos(68^{\circ} - 23^{\circ}) = \cos 45^{\circ}. \] The value of \( \cos 45^{\circ} \) is known: \[ \cos 45^{\circ} = \frac{1}{\sqrt{2}}. \] Therefore, we conclude that \[ \cos 23^{\circ} \cos 68^{\circ} + \sin 23^{\circ} \sin 68^{\circ} = \frac{1}{\sqrt{2}}. \] Thus, the final answer is \[ \boxed{\frac{1}{\sqrt{2}}}. \]