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Trigonometric equation

Source: 2007 Bulgarian Autumn Math Competition, Problem 11.1

March 17, 2022

algebraTrigonometric Equations

Problem Statement

Let

0<\alpha,\beta<\frac{\pi}{2}

which satisfy

(\cos^2\alpha+\cos^2\beta)(1+\tan\alpha\tan\beta)=2

Prove that

\alpha+\beta=\frac{\pi}{2}