Trig Fractions help...................!

1/tanx + sinx/cosx-1

Simplify help please! 10 points

1/tanx + sinx/ (cosx-1)

= cot x + sin x / (cos x -1 ) * (cos x +1) / ( cos x +1)

= cos x / sin x + sin x ( cos x +1) / (cos^2 x - 1)

= cos x / sin x + sin x ( cos x +1) / ( cos^2 x - sin^2 x - cos^2 x)

= cos x / sin x + sin x (cos x +1) / - sin^2 x

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

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

= - 1 / sin x

= - csc x