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Trigonometry › Trigonometric functions: period and extremes › Maximum and minimum

Maximum of tan beta under a constraint

High school · ●●●○○ 50 pts

Given α,β∈(0,π2)\alpha, \beta \in\left(0, \frac{\pi}{2}\right) with α+β≠π2\alpha+\beta \neq \frac{\pi}{2} and sin⁡β=2cos⁡(α+β)sin⁡α\sin \beta=2 \cos (\alpha+\beta) \sin \alpha, find the maximum value of tan⁡β\tan \beta. Round to two decimal places.

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trigonometric identitiesangle addition formulatangent addition formulaAM-GM inequalityoptimization