Question

Prove the identity. (cos^(2)x)/((1-sinx)^(2))=(1+sinx)/(1-sinx)

Answer 1

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.