If x is defined by x = tan^2 A − sin^2 A, then simplify x to an equivalent expression in terms of tan A and sin A (or a single simpler product), assuming the expressions are defined.

Introduction / Context:
This problem tests identity-based simplification with trigonometric squares. The expression tan^2 A - sin^2 A looks like a difference, but it can be factorized using the relationship between tan and sin through cos. Specifically, tan^2 A = sin^2 A / cos^2 A. Substituting that converts everything into sin and cos, after which a common factor sin^2 A can be taken out. Finally, the remaining bracket simplifies using 1 - cos^2 A = sin^2 A. The end result becomes a clean product form, which is a typical goal in simplification questions.

Given Data / Assumptions:

• x = tan^2 A - sin^2 A • Use identities: tan^2 A = sin^2 A / cos^2 A and 1 - cos^2 A = sin^2 A • Assume cos A ≠ 0 where needed

Concept / Approach:
Rewrite tan^2 A in terms of sin and cos: tan^2 A = sin^2 A / cos^2 A. Then: x = (sin^2 A / cos^2 A) - sin^2 A. Factor sin^2 A and simplify the bracket: (1/cos^2 A) - 1 = (1 - cos^2 A)/cos^2 A = sin^2 A / cos^2 A = tan^2 A. So x becomes sin^2 A * tan^2 A.

Step-by-Step Solution:
1) Start: x = tan^2 A - sin^2 A 2) Replace tan^2 A with sin^2 A / cos^2 A: x = (sin^2 A / cos^2 A) - sin^2 A 3) Factor out sin^2 A: x = sin^2 A * (1/cos^2 A - 1) 4) Combine terms inside the bracket: 1/cos^2 A - 1 = (1 - cos^2 A)/cos^2 A 5) Use identity 1 - cos^2 A = sin^2 A: (1 - cos^2 A)/cos^2 A = sin^2 A / cos^2 A 6) Recognize sin^2 A / cos^2 A = tan^2 A: x = sin^2 A * tan^2 A

Verification / Alternative check:
Take A = 45°: tan 45° = 1 and sin 45° = √2/2, so sin^2 45° = 1/2. Original: tan^2 A - sin^2 A = 1 - 1/2 = 1/2. Product form: tan^2 A sin^2 A = 1 * 1/2 = 1/2. They match, confirming the identity simplification.

Why Other Options Are Wrong:
• tan A sin A is missing squares and will not match for general A. • cot and cosec forms are reciprocals and do not represent the same difference. • sin^2 A / tan^2 A equals sin^2 A * cot^2 A, which is different from tan^2 A * sin^2 A.

Common Pitfalls:
• Forgetting that tan^2 A involves cos^2 A in the denominator. • Replacing 1 - cos^2 A incorrectly (it equals sin^2 A, not sin A). • Cancelling terms across subtraction before factoring (not valid).

Final Answer:
tan^2 A sin^2 A