In trigonometry, simplify the expression sec^6 A − tan^6 A − 3 sec^2 A tan^2 A for an acute angle A and determine its exact numerical value.

Introduction / Context:
This problem tests knowledge of trigonometric identities and algebraic factorisation. The expression sec^6 A − tan^6 A − 3 sec^2 A tan^2 A may look complicated, but it is built around the familiar identity sec^2 A − tan^2 A = 1. Recognising patterns similar to algebraic identities such as a^3 − b^3 and using basic trigonometric relations leads to a simple numerical result. These types of questions appear frequently in competitive exams under the simplification or trigonometry section.

Given Data / Assumptions:

• A is an acute angle, so sec A and tan A are well defined.
• The expression to simplify is sec^6 A − tan^6 A − 3 sec^2 A tan^2 A.
• Standard trigonometric identities such as sec^2 A − tan^2 A = 1 and sec^2 A = 1 + tan^2 A can be used.
• The final answer is expected to be a simple number or a very simple trigonometric term.

Concept / Approach:
First, treat sec^2 A and tan^2 A as algebraic variables. Rewrite sec^6 A − tan^6 A as (sec^2 A)^3 − (tan^2 A)^3 and use the factorisation for a^3 − b^3. Then simplify the resulting terms using the identity sec^2 A − tan^2 A = 1. After this, combine the remaining terms with the − 3 sec^2 A tan^2 A that appears separately. The aim is to reach a compact form that can be evaluated using basic identities without substituting any specific angle.

Step-by-Step Solution:
Step 1: Let p = sec^2 A and q = tan^2 A for simplicity.Step 2: Then sec^6 A − tan^6 A becomes p^3 − q^3.Step 3: Use a^3 − b^3 = (a − b)(a^2 + ab + b^2). Hence p^3 − q^3 = (p − q)(p^2 + pq + q^2).Step 4: We know that p − q = sec^2 A − tan^2 A = 1.Step 5: Therefore sec^6 A − tan^6 A = 1 × (p^2 + pq + q^2) = p^2 + pq + q^2.Step 6: Now write the full expression: sec^6 A − tan^6 A − 3 sec^2 A tan^2 A = (p^2 + pq + q^2) − 3pq.Step 7: Combine terms: p^2 + pq + q^2 − 3pq = p^2 − 2pq + q^2.Step 8: Recognise another algebraic identity: p^2 − 2pq + q^2 = (p − q)^2.Step 9: Substitute back p − q = 1 to get (p − q)^2 = 1^2 = 1.Step 10: Therefore the whole expression simplifies to 1, independent of the actual value of A.

Verification / Alternative check:
To verify, pick a simple angle such as A = 45°. Then sec 45° = √2 and tan 45° = 1. Compute sec^2 A = 2 and tan^2 A = 1. Then sec^6 A − tan^6 A − 3 sec^2 A tan^2 A = 2^3 − 1^3 − 3(2)(1) = 8 − 1 − 6 = 1. This confirms our algebraic result. Any other convenient angle within the domain of the functions will give the same value.

Why Other Options Are Wrong:
The options −1 and 0 are incorrect because both algebraic simplification and numerical checking show the expression is strictly positive and equal to 1. The option sec A tan A depends on A, while our result is independent of the angle. The additional option sec^2 A − tan^2 A evaluates to 1, but this is the square root of our final squared expression, so it would match only by coincidence if misapplied. The correct and exact value is 1.

Common Pitfalls:
Common mistakes include forgetting the identity sec^2 A − tan^2 A = 1, or trying to expand everything numerically without recognising the a^3 − b^3 pattern. Some learners incorrectly apply sec^2 A = 1 + tan^2 A at the wrong stage, creating unnecessary complexity. Keeping the expression in terms of sec^2 A and tan^2 A and systematically using algebraic identities makes the problem straightforward.

Final Answer:
The simplified value of sec^6 A − tan^6 A − 3 sec^2 A tan^2 A is 1.