1. The problem states: Given $f(x) = \cos x$, solve the expression $A = \frac{f(30) + f(60)}{f(\pi - 30) - f(\pi + 60)}$. 2. Substitute $f(x) = \cos x$ into the expression: $$A = \frac{\cos 30 + \cos 60}{\cos(\pi - 30) - \cos(\pi + 60)}$$ 3. Recall the cosine values in degrees (or convert to radians): - $\cos 30^\circ = \frac{\sqrt{3}}{2}$ - $\cos 60^\circ = \frac{1}{2}$ 4. Use the cosine identity for angles involving $\pi$: $$\cos(\pi - \theta) = -\cos \theta$$ $$\cos(\pi + \theta) = -\cos \theta$$ 5. Apply these identities: $$\cos(\pi - 30) = -\cos 30 = -\frac{\sqrt{3}}{2}$$ $$\cos(\pi + 60) = -\cos 60 = -\frac{1}{2}$$ 6. Substitute these values back into $A$: $$A = \frac{\frac{\sqrt{3}}{2} + \frac{1}{2}}{-\frac{\sqrt{3}}{2} - \frac{1}{2}}$$ 7. Simplify numerator and denominator: $$A = \frac{\frac{\sqrt{3} + 1}{2}}{-\frac{\sqrt{3} + 1}{2}}$$ 8. Cancel the common factor $\frac{\sqrt{3} + 1}{2}$: $$A = \frac{\cancel{\frac{\sqrt{3} + 1}{2}}}{-\cancel{\frac{\sqrt{3} + 1}{2}}} = -1$$ 9. Final answer: $$\boxed{A = -1}$$