For `x,yepsilonR` with `0 lt x lt (pi)/2` such that `((sinx)^(2y))/((cosx)^((y^(2))/2))+((cosx)^(2y))/((sinx)^((y^(2))/2))=sin2x`, then `y` is ________.

Answer

2

Explanation

Apply `AMgeGm`
`((sinx)^(2y))/((cosx)^((y^(2))/2))+((cosx)^(2y))/((sinx)^((y^(2))/2))=2(sinx.cosx)^(y-(y^(2))/4)`
It follows that `sin 2x ge 2(sinx.cosx)^(y-(y^(2))/4)`
`:' sin x .cos xlt 1implies1 le y - (y^(2))/4` or `(1-(y^(2))/2)^(2) le 0`
`impliesy=2` and `sin=cosx`
So there is a unique solution `x-(pi)/4, y=2`