How do you evaluate cot(arcsin(x-1))cot(arcsin(x−1)) without a calculator?

1 Answer
A. S. Adikesavan
Nov 27, 2016

sqrt(2x-x^2)/(x-1), x in [0, 2]√2x−x2x−1,x∈[0,2]

Explanation:

Let a =arc sin ( x - 1 ) in [-pi/2, pi/2]a=arcsin(x−1)∈[−π2,π2].

Then sin a = x-1 in [-1, 1] to x in [0, 2]sina=x−1∈[−1,1]→x∈[0,2].

The given expression

cot a = cos a/sin a=sqrt (1-sin^2a)/sin a=sqrt(1-(x-1)^2)/(x-1)=sqrt(2x-x^2)/(x-1)cota=cosasina=√1−sin2asina=√1−(x−1)2x−1=√2x−x2x−1

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