Use a sum or difference formula to find the exact value of the trigonometric function. \( \cos 75^{\circ} \) \( \cos 75^{\circ}= \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

To find the exact value of \( \cos 75^{\circ} \) using the sum formula, we can express \( 75^{\circ} \) as the sum of \( 45^{\circ} \) and \( 30^{\circ} \): \[ 75^{\circ} = 45^{\circ} + 30^{\circ} \] Now, apply the cosine sum formula: \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \] Substituting \( A = 45^{\circ} \) and \( B = 30^{\circ} \): \[ \cos 75^{\circ} = \cos 45^{\circ} \cos 30^{\circ} - \sin 45^{\circ} \sin 30^{\circ} \] We know the exact values of these trigonometric functions: \[ \cos 45^{\circ} = \frac{\sqrt{2}}{2}, \quad \cos 30^{\circ} = \frac{\sqrt{3}}{2}, \quad \sin 45^{\circ} = \frac{\sqrt{2}}{2}, \quad \sin 30^{\circ} = \frac{1}{2} \] Plugging these values in: \[ \cos 75^{\circ} = \left( \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} \right) - \left( \frac{\sqrt{2}}{2} \times \frac{1}{2} \right) \] Simplify the expression: \[ \cos 75^{\circ} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \] **Final Answer:** \[ \cos 75^{\circ} = \frac{\,\sqrt{6}\ -\ \sqrt{2}\,}{\,4\,} \]