Given Equation:

(1-sinx)² * (1-sinx)² / 2cos²x = 1sin²x / 1-sin²x

To simplify the given equation, we will start by simplifying each side separately.

Simplifying the Left Side:

(1-sinx)² * (1-sinx)² / 2cos²x

Step 1: Rewrite (1-sinx)² as (1-sinx)(1-sinx)

(1-sinx)(1-sinx) * (1-sinx)(1-sinx) / 2cos²x

Step 2: Expand the numerator using the distributive property

(1-sinx)(1-sinx) = 1 - sinx - sinx + sin²x = 1 - 2sinx + sin²x

(1 - 2sinx + sin²x)(1 - 2sinx + sin²x) / 2cos²x

Step 3: Simplify the numerator by multiplying the binomials

(1 - 2sinx + sin²x)² / 2cos²x

Step 4: Square each term in the numerator

(1 - 4sinx + 4sin²x - 2sinx + 4sin²x - 2sin²x + sin⁴x) / 2cos²x

Step 5: Combine like terms in the numerator

(1 - 6sinx + 7sin²x - 2sin²x + sin⁴x) / 2cos²x

Step 6: Simplify the expression by combining the remaining terms

(1 - 6sinx - sin²x + sin⁴x) / 2cos²x

Simplifying the Right Side:

1sin²x / 1-sin²x

Step 1: Simplify the numerator

sin²x / 1-sin²x

Step 2: Rewrite sin²x as (1-cos²x)

(1-cos²x) / 1-sin²x

Step 3: Rewrite 1-sin²x as cos²x

(1-cos²x) / cos²x

Step 4: Use the property of reciprocals to simplify the expression

1 / cos²x - cos²x / cos²x

Step 5: Simplify the expression

1 / cos²x - 1

Final Simplified Equation:

(1 - 6sinx - sin²x + sin⁴x) / 2cos²x = 1 / cos²x - 1

Conclusion:

The given equation (1-sinx)² * (1-sinx)² / 2cos²x = 1sin²x / 1-sin²x simplifies to (1 - 6sinx - sin²x + sin⁴x) / 2cos²x = 1 / cos²x - 1.

both sides are equivalent how you can tell me to solve this for x