Use the Half-angle Formulas to find the exact value of the expression \( \cos 105^{\circ} \) \( \cos 105^{\circ}=\square \) (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

To find \( \cos 105^{\circ} \), we can use the half-angle formula. First, we recognize that \( 105^{\circ} \) is half of \( 210^{\circ} \). Therefore, we can use the half-angle formula for cosine: \[ \cos \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}} \] Here, \( \theta = 210^{\circ} \). First, we need to calculate \( \cos 210^{\circ} \): \[ \cos 210^{\circ} = -\frac{\sqrt{3}}{2} \] Now, substituting this value into the half-angle formula: \[ \cos 105^{\circ} = \cos \left(\frac{210^{\circ}}{2}\right) = \sqrt{\frac{1 + (-\frac{\sqrt{3}}{2})}{2}} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2} \] Thus, the exact value is: \[ \cos 105^{\circ} = \frac{\sqrt{2 - \sqrt{3}}}{2} \]