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My contention is that you cannot apply the Inverse Function Theorem to this problem because there is a point in the interval at which the derivative is zero.
At x = pi/2, f ' = 0.
http://i111.photobucket.com/albums/n149/camarolt4z28/Untitled.png
At x = pi/2, f ' = 0.
http://i111.photobucket.com/albums/n149/camarolt4z28/Untitled.png