Using the Integral Mean Value Theorem

Approach: Since $\cos x$ is continuous on $[0,2]$, the integral mean value theorem guarantees some $\xi\in[0,2]$ with $\int_0^2 \cos x \, dx = \cos \xi \cdot (2-0)$. Pin down $\cos \xi$ by evaluating the integral exactly and dividing by the length of the interval.

Steps:

1. Apply the theorem: $\int_0^2 \cos x \, dx = \cos \xi \cdot 2$ for some $\xi\in[0,2]$.
2. Evaluate the true value: $[\sin x]_0^2 = \sin 2 \approx 0.909$.
3. Divide by the interval length: $\cos\xi = \tfrac{\sin 2}{2} \approx 0.4546$.

Answer: There exists $\xi\in[0,2]$ with $\cos\xi \approx 0.4546$; the integral is approximately $0.909$.