Here is the solution to the problem:
Let's analyze each option:
A) cosec²A * cos²A = cos²A / sin²A
For this equation to be true, both sides of the equation must equal. Here, the left side is the reciprocal of sin²A (cosec²A) times cos²A and the right side is cos²A divided by sin²A. When we simplify the left side, both sides match and hence this identity is correct.
B) tan²A * sec²A = sin²A / cos²A
This equation is incorrect. Although tan²A equals sin²A / cos²A, multiplying tan²A by sec²A is equivalent to multiplying it by 1 / cosA which changes the left side of the equation. Hence, this identity is false.
C) cot²A * cos²A = sin²A / cos²A
This one is also incorrect. Cotangent squared A is equal to cos²A / sin²A, and multiplying this by cos²A alters the equation. This identity proves to be false.
D) sec²A * csc²A = tan²A / sin²A
This equation is incorrect. The left side of the equation is the product of the reciprocal of cos²A and the reciprocal of sin²A which is not the same as the tan²A divided by sin²A. Hence this identity is false.
Therefore, only the identity A) cosec²A * cos²A = cos²A / sin²A is true.