1. **State the problem:** Simplify the expression $\cos A + \sin A$. 2. **Recall the formula:** There is no direct simplification for $\cos A + \sin A$ alone, but it can be expressed as a single trigonometric function using the identity: $$\cos A + \sin A = \sqrt{2} \sin\left(A + \frac{\pi}{4}\right) = \sqrt{2} \cos\left(A - \frac{\pi}{4}\right)$$ 3. **Explanation:** This comes from the sum-to-product formulas and the fact that $\sin x$ and $\cos x$ are phase-shifted versions of each other. The amplitude is $\sqrt{1^2 + 1^2} = \sqrt{2}$. 4. **Derivation:** $$\cos A + \sin A = \sqrt{2} \left( \frac{1}{\sqrt{2}} \cos A + \frac{1}{\sqrt{2}} \sin A \right) = \sqrt{2} \left( \cos \frac{\pi}{4} \cos A + \sin \frac{\pi}{4} \sin A \right)$$ Using the cosine addition formula: $$\cos x \cos y + \sin x \sin y = \cos(x - y)$$ So, $$\cos A + \sin A = \sqrt{2} \cos\left(A - \frac{\pi}{4}\right)$$ 5. **Final answer:** $$\boxed{\cos A + \sin A = \sqrt{2} \cos\left(A - \frac{\pi}{4}\right)}$$