Question

Use ley de los cosenos para demostrar la identidad:
cos α÷a+cos β÷b+cosγ÷c=a²+b²+c²÷2abc

Answer 1

Por ley de cosenos tenemos

$\cos \alpha = \dfrac{b^2+c^2-a^2}{2bc}\to \dfrac{\cos \alpha}{a}= \dfrac{b^2+c^2-a^2}{2abc}\\ \\ \\ \cos \beta = \dfrac{a^2+c^2-b^2}{2ac}\to \dfrac{\cos \beta}{b}= \dfrac{a^2+c^2-b^2}{2abc}\\ \\ \\ \cos \theta = \dfrac{a^2+b^2-c^2}{2ab}\to \dfrac{\cos \theta}{c}= \dfrac{a^2+b^2-c^2}{2abc}\\ \\ \\ \texttt{por ende: } \dfrac{\cos \alpha}{a}+\dfrac{\cos \beta}{b}+\dfrac{\cos \theta}{c}= \dfrac{a^2+b^2+c^2}{2abc}$