The trigonometric identity `cos x/(1-tanx) + sin x/(1-sinx) = sin x + cos x` has to be  proved.

If x = 45 degrees, the left hand side is equal to `oo` as `(1/sqrt 2)/(1 - 1) = 1/0 = oo` , but the right hand side is equal to `sqrt 2`

The given equation is not a trigonometric identity.

If the `1 - sin x` in the term `sin x/(1 - sin x)` is replaced with `1 - cot x` we have:

`cos x/(1-tanx) + sin x/(1-cot x)`

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

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

= `(cos^2 x - sin^2x)/(cos x - sin x)`

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

= `cos x + sin x`

The equation `cos x/(1-tanx) + sin x/(1-sinx) = sin x + cos x` is not a trigonometric identity. `cos x/(1-tanx) + sin x/(1-cot x) = sin x + cos x` on the other hand is a trigonometric identity.