1. The problem is to simplify or analyze the expression $\sin x + \cos x$. 2. A useful formula to rewrite sums of sine and cosine is the amplitude-phase form: $$\sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)$$ This comes from the identity $a \sin x + b \cos x = \sqrt{a^2 + b^2} \sin(x + \phi)$ where $\phi = \arctan\left(\frac{b}{a}\right)$. 3. Here, $a = 1$ and $b = 1$, so: $$\sqrt{1^2 + 1^2} = \sqrt{2}$$ and $$\phi = \arctan\left(\frac{1}{1}\right) = \frac{\pi}{4}$$ 4. Therefore, the expression simplifies to: $$\sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)$$ 5. This form is useful for understanding the amplitude and phase shift of the combined wave. Final answer: $$\sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)$$