Maximul unei expresii trigonometrice

[Mateescu Constantin]

Fie \( a_1,\ a_2,\ ...,\ a_n \in \left\(0,\ \frac{\pi}{2}\right\)\ ,\ \ n\in \mathbb{N}, \ n\ge 2 \), astfel incat \( \prod_{k=1}^{n}\tan\ a_k=1 \).
Sa se determine maximul expresiei:\( \prod_{k=1}^{n}\sin\ a_k \).

G.M. 4/2002

[Marius Mainea]

Folosind AM-GM

\( \prod \sin a_k=\prod\frac{\tan a_k}{\sqrt{1+\tan^2a_k}}\le \frac{1}{\sqrt{\prod 2\tan a_k}}=2^{-\frac{n}{2}} \)