We have to find the Minimum and maximum value of $$P= {{\sqrt 3} + \sin A + \sin B + \sin C } \over { 2 {\sin A}{\sin B}{\sin C}} $$ using Lagrange multipliers , where ABC is a triangle .
I know the solution using $\sin {A} = {a \over {2R}} $ and other trigonometric properties . This question has already been asked on the forum , but someone mentioned using Lagrange multiplier would be a much easier and helpful way .
But myself have not been able to do so ,so please post a solution using Lagrange multipliers if possible .
My attempt so I try to express $\sin C$ in terms of $\sin A$ and $\sin B $
$$\sin C = \sin ({{\pi - { (A +B)}}}) = \sin {( A +B )} = \sin A{ \sqrt{ 1- \sin^2 B } + { \sin B}{ \sqrt { 1 - \sin^2 A}} } $$
Now I don't know what to do further .