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