Inegalitate trigonometrica

[Adriana Nistor]

Aratati ca in orice triunghi ascutitunghic are loc inegalitatea:
\( \frac{b+c}{cos A} + \frac{a+c}{cos B} + \frac{a+b}{cos C} \ge 4(a+b+c) \)

[BogdanCNFB]

SOLUTIE:
Din T.Cosinusului avem ca \( a=b\cdot\cos C+c\cdot\cos B;b=c\cdot\cos A+a\cdot\cos C;c=a\cdot\cos B+b\cdot\cos A \)
\( LHS=\sum(\frac{a}{\cos B}+\frac{b}{\cos A})=\sum(\frac{a^2}{a\cdot\cos B}+\frac{b^2}{b\cdot\cos A})\ge \sum\frac{(a+b)^2}{c}\ge\fra{(2\sum a)^2}{\sum a}=RHS \).