Problem 1

Determine all values of

x

in the interval

0 \le x \le 2\\pi

which satisfy

2\\cos x \le |\\sqrt{1+\\sin 2x}-\\sqrt{1-\\sin 2x}| \le \\sqrt{2}

Step 3 of 4: Test the nonnegative-cosine intervals

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\\cos 2x\\ge0:\\quad x\\in[0,\\pi/4]\\cup[3\\pi/4,5\\pi/4]\\cup[7\\pi/4,2\\pi].

Detailed analysis

Here the first inequality is $\\cos x\\le|\\sin x|$ . It holds only at the endpoint $\\pi/4$ in the first interval and at $7\\pi/4$ in the last; it holds throughout $[3\\pi/4,5\\pi/4]$ . The upper bound $2|\\sin x|\\le\\sqrt2$ also holds throughout that middle interval.