Final answer:
To prove the identity (cos^2x) / ((1 - sinx)^2) = (1 + sinx) / (1 - sinx), we can use the trigonometric identity sin^2x + cos^2x = 1 and simplify both sides of the equation.
Step-by-step explanation:
To prove the identity (cos2x) / ((1 - sinx)2) = (1 + sinx) / (1 - sinx), we need to start with the left side of the equation and manipulate it to obtain the right side.
1. Start with the left side: (cos2x) / ((1 - sinx)2)
2. Use the identity sin2x + cos2x = 1 to replace cos2x with 1 - sin2x:
(1 - sin2x) / ((1 - sinx)2)
3. Simplify the numerator: (1 - sin2x) = cos2x
4. Simplify the denominator: ((1 - sinx)2) = (1 - 2sinx + sin2x)
5. Substitute the simplified numerator and denominator back into the equation:
cos2x / (1 - 2sinx + sin2x)
6. Factor the denominator: (1 - sinx)(1 - sinx)
7. Cancel out the common factors in the numerator and denominator:
cos2x / (1 - sinx)
8. Rearrange the numerator and denominator: (1 + sinx) / (1 - sinx)
9. This is equal to the right side of the equation.