A Cosine Substitution Bound

★ 9/10trigonometry Substitution Trigonometric Identities inequality

Problem

Positive reals $x, y, z$ satisfy

x^2 + y^2 + z^2 + 2xyz = 1.

Prove that

\sqrt{\frac{1-x}{1+x}} + \sqrt{\frac{1-y}{1+y}} + \sqrt{\frac{1-z}{1+z}} \geqslant \sqrt{3}.

Answer

Solution

Difficulty	9/10
Topics	trigonometry, Substitution, Trigonometric Identities, inequality

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