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 4 of 4: Test the negative-cosine intervals

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\\cos 2x<0:\\quad x\\in(\\pi/4,3\\pi/4)\\cup(5\\pi/4,7\\pi/4),\\quad 2|\\cos x|\\le\\sqrt2.

Detailed analysis

The lower inequality is automatic because $\\cos x\\le|\\cos x|$ . On both intervals, $|\\cos x|<1/\\sqrt2$ , so the upper inequality holds. Combining these intervals with the middle interval from the previous case gives $x\\in[\\pi/4,7\\pi/4]$ .