1. **State the problem:** Find the exact value of $\cos 75^\circ$ using the formula for the cosine of a sum of angles. 2. **Recall the formula:** $$\cos(u + v) = \cos u \cos v - \sin u \sin v$$ This formula allows us to express the cosine of a sum of two angles in terms of the sines and cosines of the individual angles. 3. **Choose angles $u$ and $v$ such that $u + v = 75^\circ$:** A convenient choice is $u = 45^\circ$ and $v = 30^\circ$ because their sine and cosine values are well known. 4. **Apply the formula:** $$\cos 75^\circ = \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ$$ 5. **Substitute known values:** $$\cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 30^\circ = \frac{1}{2}$$ 6. **Calculate:** $$\cos 75^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}$$ 7. **Simplify the expression:** Multiply numerator and denominator by $\sqrt{2}$ to rationalize the denominator: $$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{12} - 2}{4 \sqrt{2}} = \frac{2\sqrt{3} - 2}{4 \sqrt{2}} = \frac{2(\sqrt{3} - 1)}{4 \sqrt{2}} = \frac{\sqrt{3} - 1}{2 \sqrt{2}}$$ 8. **Final answer:** $$\cos 75^\circ = \frac{\sqrt{3} - 1}{2 \sqrt{2}}$$ This matches option A. **Answer: A) $\frac{\sqrt{3} - 1}{2 \sqrt{2}}$**