1. The problem is to calculate $\cos 75^\circ$ and round the result to 2 decimal places. 2. We can use the angle sum identity for cosine: $$\cos(a+b) = \cos a \cos b - \sin a \sin b$$ 3. Express $75^\circ$ as $45^\circ + 30^\circ$. 4. Substitute into the identity: $$\cos 75^\circ = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ$$ 5. Use known exact 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. Substitute these values: $$\cos 75^\circ = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}$$ 7. Calculate the numerical value: $$\sqrt{6} \approx 2.449, \quad \sqrt{2} \approx 1.414$$ $$\cos 75^\circ \approx \frac{2.449 - 1.414}{4} = \frac{1.035}{4} = 0.25875$$ 8. Round to 2 decimal places: $$\cos 75^\circ \approx 0.26$$ Final answer: $\boxed{0.26}$