If \sin \alpha+\cos \alpha=\sqrt{2}, where 0\lt \alpha\lt \frac{\pi}{2}, then what is \sin^{3}\alpha-\cos^{3}\alpha equal to?

Explanation

Given \sin \alpha + \cos \alpha = \sqrt{2}, dividing by \sqrt{2} gives \sin(\alpha+45^\circ)=1, so \alpha = 45^\circ. Thus \sin 45^\circ = \cos 45^\circ, making \sin^3 \alpha - \cos^3 \alpha = 0.

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