cot x = cosx/sinx
cosx(cosx/sinx)/1-sinx - 1 = cscx
cos^2x/sinx(1-sinx) - 1 = cscx
cos^2x/sinx(1-sinx) - sinx(1-sinx)/sinx(1-sinx) = cscx
[cos^2x - sinx(1-sinx)] / sinx(1-sinx) = csc x
[cos^2x - sinx + sin^2x]/ sinx(1-sinx) = csc x
but cos^2x + sin^2x = 1
1 - sinx/sinx(1-sinx) = cscx
1 - sinx cancels each other we have;
1/sinx =cscx
and cosec x is equal to 1 / sine x trigonometrically