Now, since \(0 < x < \frac{\pi}{2}\), \(u = \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)\), so \(u \in (1, \sqrt{2}]\) (maximum \(\sqrt{2}\) at \(x = \frac{\pi}{4}\)).

["Optimizing ( u = \sin x + \cos x ) for ( 0 < x < \frac{\pi}{2} ):\nMaximize ( u \in (1, \sqrt{2}] ) when ( x \in (0, \frac{\pi}{2}) )", "When analyzing trigonometric expressions involving sine and cosine, few identities are as elegant and practical as transforming ( u = \sin x + \cos x ) into a single sine function with a phase shift. For ( 0 < x < \frac{\pi}{2} ), this transformation reveals powerful insights about the range and maximum value of ( u ).", "### Step 1: Rewrite ( u ) Using a Trigonometric Identity\nThe key identity is:\n[\nu = \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)\n]\nThis transformation leverages the amplitude-scaled sine function. By writing ( u ) this way, we shift and scale the basic sine wave to better reflect the original expression’s behavior.", "### Step 2: Analyze the Range Based on the Identity\nSince ( x \in \left(0, \frac{\pi}{2}\right) ), the phase-shifted angle satisfies:\n[\nx + \frac{\pi}{4} \in \left(\frac{\pi}{4}, \frac{3\pi}{4}\right)\n]\nWithin this interval, the sine function achieves its maximum value on the unit circle at ( \frac{\pi}{2} ). Thus:\n[\n\sin\left(x + \frac{\pi}{4}\right) \in \left(\frac{\sqrt{2}}{2}, 1\right]\n]\nMultiplying by ( \sqrt{2} ), we find:\n[\nu = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \in \left(1, \sqrt{2}\right]\n]\nThe strict inequality at the lower bound (1) occurs as ( x \ o 0^+ ), and the maximum value ( \sqrt{2} ) is attained precisely when ( x = \frac{\pi}{4} ), because:\n[\n\sin\left(\frac{\pi}{4} + \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{2}\right) = 1\n]", "### Why This Range Matters\nUnderstanding the bounded nature of ( u ) is crucial in optimization problems involving trigonometric functions. The maximum value ( \sqrt{2} ) makes this expression particularly valuable in applications such as signal processing, harmonic motion, and geometry where ( \sin x + \cos x ) arises naturally.", "### Conclusion\nFor all ( x ) in the open interval ( \left(0, \frac{\pi}{2}\right) ), the quantity\n[\nu = \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)\n]\nlies within ( (1, \sqrt{2}] ), reaching its peak of ( \sqrt{2} ) at ( x = \frac{\pi}{4} ). Mastering this identity and range helps simplify complex trigonometric maximization tasks into intuitive, scalable computations.", "Keywords:\n( \sin x + \cos x ), ( u = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) ), maximum value ( \sqrt{2} ), interval ( 0 < x < \frac{\pi}{2} ), trigonometric identity, function analysis, optimization trigonometry", "Meta description:\nExplore how ( u = \sin x + \cos x ) simplifies to ( \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) ) for ( x \in (0, \frac{\pi}{2}) ), achieving maximum ( \sqrt{2} ) at ( x = \frac{\pi}{4} ). Learn the ideal range and applications.", "---", "Discover more about trigonometric function identities and maxima in our expanded guide to maximizing ( \sin x + \cos x ) across key intervals."]