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

Nov 27, 2016

$\frac{\sqrt{2 x - {x}^{2}}}{x - 1} , x \in \left[0 , 2\right]$

Explanation:

Let $a = a r c \sin \left(x - 1\right) \in \left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$.

Then $\sin a = x - 1 \in \left[- 1 , 1\right] \to x \in \left[0 , 2\right]$.

The given expression

$\cot a = \cos \frac{a}{\sin} a = \frac{\sqrt{1 - {\sin}^{2} a}}{\sin} a = \frac{\sqrt{1 - {\left(x - 1\right)}^{2}}}{x - 1} = \frac{\sqrt{2 x - {x}^{2}}}{x - 1}$