1-cos2x-sinx/cosx-sin2x

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To simplify this expression, we will first rewrite it as:

(1 - cos^2x - sinx) / (cosx - sin^2x)

Next, we will use trigonometric identities such as sin^2x + cos^2x = 1 and the double angle identities to simplify the expression.

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

Now we can factor the numerator using the difference of squares identity:

= ((sinx + cosx)(sinx - cosx) - sinx) / (cos^2x + cosx - 1)
= (sinxsinx - cosxcosx - sinx) / (cos^2x + cosx - 1)
= (sin2x - sinx) / (cos^2x + cosx - 1)

Now, we can simplify the expression further:

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

Therefore, (1 - cos^2x - sinx) / (cosx - sin^2x) simplifies to -cscx.