Final answer:
The trigonometric identity provided by the student is verified using the Pythagorean trigonometric identity. By rearranging and simplifying, we find that the left-hand side of the equation simplifies to sin²(\(\theta\)), which is indeed the reciprocal of csc²(\(\theta\)). Therefore, the identity is correct and corresponds to answer A) sin²(\(\theta\)).
Step-by-step explanation:
The trigonometric identity in question can be verified by recognizing a basic Pythagorean trigonometric identity and then simplifying the expression accordingly. The identity to use is sin²(\(\theta\)) + cos²(\(\theta\)) = 1, which can be rearranged to express sin²(\(\theta\)) as 1 - cos²(\(\theta\)).
Now, let's verify the identity: ((1-cos²(\(\theta\)))(1+cos²(\(\theta\))) = csc²(\(\theta\)). First, expand the left-hand side to get 1 - cos²(\(\theta\)) - cos²(\(\theta\)) + cos´(\(\theta\)), which simplifies to 1 - 2cos²(\(\theta\)) + cos´(\(\theta\)). Since sin²(\(\theta\)) represents the same as 1 - cos²(\(\theta\)), we substitute this in to get sin²(\(\theta\)), which is indeed csc²(\(\theta\)) inverted. Therefore, the identity is verified and the answer is A) sin²(\(\theta\)).
The trigonometric identity provided by the student is verified using the Pythagorean trigonometric identity. By rearranging and simplifying, we find that the left-hand side of the equation simplifies to sin²(\(\theta\)), which is indeed the reciprocal of csc²(\(\theta\)). Therefore, the identity is correct and corresponds to answer A) sin²(\(\theta\)).