1. The problem is to calculate $\cos(75^\circ) \times 10$ and round the result to 2 decimal places. 2. Recall the cosine of an angle in degrees can be found using a calculator or trigonometric identities. Here, $75^\circ$ is not a standard angle, but can be expressed as $45^\circ + 30^\circ$. 3. Using the cosine addition formula: $$\cos(a + b) = \cos a \cos b - \sin a \sin b$$ where $a = 45^\circ$ and $b = 30^\circ$. 4. Substitute values: $$\cos 75^\circ = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ$$ 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. Calculate: $$\cos 75^\circ = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}$$ 7. Approximate numerically: $$\sqrt{6} \approx 2.4495, \quad \sqrt{2} \approx 1.4142$$ $$\cos 75^\circ \approx \frac{2.4495 - 1.4142}{4} = \frac{1.0353}{4} = 0.2588$$ 8. Multiply by 10: $$0.2588 \times 10 = 2.588$$ 9. Round to 2 decimal places: $$2.59$$ Final answer: $2.59$