Solve the equation $\displaystyle 2^{\cos x} 3^{\sin y} = z^2 - 2z + 7$. Then find the value of $x+y+z$

The minimum values of the exponential functions on the left side are 1, making the minimum of the entire left side exactly 1. By completing the square on the right side, we get (z - 1)^2 + 6, whose minimum value is also 6 when z equals 1. However, testing for the true minimum of the left side requires checking when the bases are zero, so if cos x equals 0 and sin y equals 0, the left side equals 1. Equating this to the right side gives (z - 1)^2 equal to 0, meaning z is 1, cos x is 0 so x equals pi/2, and sin y is 0 so y equals 0. Adding these values gives pi/2 plus 0 plus 1, resulting in pi/2+1.

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