1. The problem asks us to find and simplify the difference quotient $$\frac{f(a+h)-f(a)}{h}$$ for the function $$f(x) = \cos x$$. 2. Substitute the function into the difference quotient: $$\frac{\cos(a+h) - \cos(a)}{h}$$ 3. Use the cosine addition formula: $$\cos(a+h) = \cos a \cos h - \sin a \sin h$$ 4. Substitute this back: $$\frac{(\cos a \cos h - \sin a \sin h) - \cos a}{h} = \frac{\cos a \cos h - \sin a \sin h - \cos a}{h}$$ 5. Group terms with \(\cos a\): $$\frac{\cos a (\cos h - 1) - \sin a \sin h}{h}$$ 6. This is the simplified form of the difference quotient for $$f(x) = \cos x$$: $$\frac{\cos a (\cos h - 1) - \sin a \sin h}{h}$$ This expression is useful in calculus for finding the derivative of $$\cos x$$ as $$h \to 0$$.