f(x) = cot(x), (0, pi) Show that f is strictly monotonic on the given interval and therefore has an inverse function on that interval.

We are asked to determine if the function y=cot(x) on (0,pi)  has an inverse function by finding if the function is strictly monotonic on the interval using the derivative.
y'=-csc^2(x) . On (0,pi),-csc^2(x)<0 for all x so the function is monotonic and has an inverse function.
 
 

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