Prove that `[(x^2+y^2+z^2)/(x+y+z)]^(x+y+z)> x^x y^y z^z >[(x+y+z)/3]^(x+y+z)(x ,y ,z >0)`

Explanation

Let `tan^(2) alpha = x` and `tan^(2) beta = y`
`:. (sec^(4) alpha)/(tan^(2) beta) + (sec^(4) beta)/(tan^(2) alpha) = ((x + 1)^(2))/(y) + ((y + 1)^(2))/(x)`
`= (x^(2))/(y) + (y^(2))/(x) + (1)/(x) + (1)/(y) + 2 (x)/(y) + 2 (y)/(x)`
Now, using A.M `ge` G.M we get ltbegt
`((x^(2))/(y) + (y^(2))/(x) + (1)/(x) + (1)/(y) + (x)/(y) + (x)/(y) + (y)/(x) + (y)/(x))/(8) ge ((x^(2))/(y). (y^(2))/(x).(1)/(x).(1)/(y).(1)/(y).(x)/(y).(x)/(y).(y)/(x).(y)/(x))^((1)/(8))`
`implies (x^(2))/(y) + (y^(2))/(x) + (1)/(x) + (1)/(y) + (X)/(y) + (x)/(y) + (y)/(x) + (y)/(x) ge 8`
`:. (sec^(4) alpha)/(tan^(2) beta) + (sec(4) beta)/(tan^(2) alpha) ge 8`