Simplify $\tan^{-1}(\cot(a))$ to $\frac{\pi }{2}-a$ assuming $0<a<\frac{\pi }{2}$

Similar to the answer here, we can use PowerExpand.

PowerExpand[ArcTan[Cot[a]], Assumptions -> 0 < a < π/2]

π/2 - a

In fact an almost identical example appears in the PowerExpand ref page here.

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Reduce[{ArcTan[Cot[a]] - Pi/2 + a == 0, a > 0, a < Pi/2}, a, Reals]

0 < a < π/2

The answer means the inequalities a > 0, a < Pi/2 imply the relation ArcTan[Cot[a]] - Pi/2 + a == 0.

The second answer is as follows.

Simplify[D[ArcTan[Cot[x]] - Pi/2 + x, x], Assumptions -> x > 0 && x < Pi/2]

0

implies ArcTan[Cot[x]] - Pi/2 + x is a constant on the interval $(0,\pi/2)$. It remains

ArcTan[Cot[x]] - Pi/2 + x /. x -> Pi/4

0

to prove the identity.

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