Proof of cosec²A-cos²A/ cot²A = sec²A - sin²A

Step 1: Convert cot²A to cos²A/ sin²A
To simplify the left side of the equation, we need to convert cot²A to cos²A/ sin²A:
cosec²A - cos²A/ cot²A = cosec²A - cos²A/(cos²A/ sin²A) = cosec²A - sin²A

Step 2: Convert sec²A to 1/cos²A
To simplify the right side of the equation, we need to convert sec²A to 1/cos²A:
sec²A - sin²A = 1/cos²A - sin²A

Step 3: Convert sin²A to 1 - cos²A
To further simplify the right side of the equation, we need to convert sin²A to 1 - cos²A:
1/cos²A - sin²A = 1/cos²A - (1 - cos²A) = 1/cos²A - 1 + cos²A

Step 4: Combine like terms on the right side of the equation
To simplify the right side of the equation further, we need to combine like terms:
1/cos²A - 1 + cos²A = (1 - cos²A + cos⁴A)/cos²A

Step 5: Simplify the right side of the equation
To simplify the right side of the equation even further, we need to factor the numerator of the fraction:
(1 - cos²A + cos⁴A) = (1 - cos²A)(1 + cos²A) = sin²A/cos²A * (1 + cos²A) = sin²A + sin²A cos²A/cos²A = sin²A + sin²A = 2sin²A

Step 6: Simplify the left side of the equation
Now that we have simplified the right side of the equation, we can simplify the left side of the equation using the result from Step 1:
cosec²A - sin²A = (1/sin²A) - sin²A = (1 - sin⁴A)/sin²A = (1 - sin²A)(1 + sin²A)/sin²A = cos²A/sin²A = 1/cos²A

Step 7: Compare the simplified left and right sides of the equation
Now that we have simplified both sides of the equation, we can compare them:
1/cos²A = 1/cos²A

Therefore, we have proved that cosec²A-cos²A/ cot²A = sec²A - sin²A.