I've done this one multiply times yet it never seems to work out, I'm supposed to simplify yet it never works out

tan(csc^-1(x/x-1))

my first step;

sin/cos(1/sin(x/x-1))

take the messy part ...

csc^-1 (x/(x-1) ) represents an angle Ø so that

csc Ø = x/(x-1)
or
sinØ = (x-1)/x

so make a sketch of a right-angled triangle with hypotenuse x and opposite side x-1
call the adjacent side b
then b^2 + (x-1)^2 = x^2
b^2 + x^2 - 2x + 1 = x^2
b^2 = 2x - 1
b = √(2x-1) ----> the adjacent side

so we have to find tanØ

tanØ = opposite/adjacent
= (x-1)/√(2x - 1)

This approach works for most of these type of problems