MathLabs

Problem 1

Determine all values of xx in the interval 0≤x≤2pi0 \le x \le 2\\pi which satisfy 2cosx≤∣sqrt1+sin2x−sqrt1−sin2x∣≤sqrt22\\cos x \le |\\sqrt{1+\\sin 2x}-\\sqrt{1-\\sin 2x}| \le \\sqrt{2}.
Step 1 of 4: Square the radical difference
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In plain words

The square roots hide only a sign choice; squaring exposes the double-angle cosine.

y=∣sqrt1+sin2x−sqrt1−sin2x∣,qquady2=2−2sqrt1−sin22x=2−2∣cos2x∣.y=|\\sqrt{1+\\sin 2x}-\\sqrt{1-\\sin 2x}|,\\qquad y^2=2-2\\sqrt{1-\\sin^2 2x}=2-2|\\cos 2x|.
Unit circle at the first boundary angle
A unit circle marking the angle $\\pi/4$, where the sign interval changes.
Detailed analysis

Both radicands are nonnegative. Squaring the absolute difference produces the factor ∣cos2x∣|\\cos 2x|, so the remaining work is determined by the sign of cos2x\\cos 2x.