If 2\sin^{4}\alpha+2\cos^{4}\alpha-1=0, where 0\leq\alpha\lt \pi/2, then what is \sin 2\alpha+\cos 2\alpha equal to?

Explanation

2(\sin^4\alpha + \cos^4\alpha) = 1 \implies 2(1 - 2\sin^2\alpha\cos^2\alpha) = 1 \implies 2 - \sin^2 2\alpha = 1 \implies \sin^2 2\alpha = 1. Since 0\leq\alpha\lt \pi/2, \sin 2\alpha = 1, which means 2\alpha = \pi/2 so \cos 2\alpha = 0. Sum is 1+0=1.

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