In trigonometric simplification, evaluate the algebraic trigonometric product: (sec A + cos A) (sec A − cos A). Express the result in its simplest form using basic trigonometric identities.

Introduction / Context:
This question focuses on simplification of a trigonometric expression that looks similar to the algebraic form (a + b)(a − b) = a^2 − b^2. The goal is to rewrite the given product in terms of simpler trigonometric functions and then match it to one of the given options. Problems like this are common in trigonometry sections of aptitude tests because they test both algebraic pattern recognition and understanding of trigonometric identities.

Given Data / Assumptions:
- The expression to simplify is (sec A + cos A)(sec A − cos A).
- All angles are within a domain where sec A and cos A are defined, and cos A is non zero (so sec A exists).
- We should use known identities such as sec A = 1 / cos A and sec^2 A = 1 + tan^2 A.

Concept / Approach:
First observe that (sec A + cos A)(sec A − cos A) has the form (u + v)(u − v) = u^2 − v^2. Therefore, the product simplifies to sec^2 A − cos^2 A. Then we express sec^2 A as 1 / cos^2 A and perform algebraic simplification. The final result can be transformed into an equivalent expression involving sin^2 A and tan^2 A, which can then be matched with one of the answer choices.

Step-by-Step Solution:
1) Recognise a difference of squares: (sec A + cos A)(sec A − cos A) = sec^2 A − cos^2 A. 2) Replace sec^2 A with 1 / cos^2 A, giving 1 / cos^2 A − cos^2 A. 3) Write this as a single fraction: (1 − cos^4 A) / cos^2 A. 4) Factor the numerator: 1 − cos^4 A = (1 − cos^2 A)(1 + cos^2 A). 5) Use 1 − cos^2 A = sin^2 A. The expression becomes sin^2 A (1 + cos^2 A) / cos^2 A. 6) Split the fraction: sin^2 A / cos^2 A + sin^2 A. 7) Recognise sin^2 A / cos^2 A as tan^2 A, so the final simplified form is tan^2 A + sin^2 A.

Verification / Alternative check:
Take a simple angle where both sine and cosine have known values, such as A = 45 degrees. Then sec 45° = √2 and cos 45° = √2 / 2. The original product becomes (√2 + √2/2)(√2 − √2/2). Evaluating this numerically gives a certain value. Computing sin^2 A + tan^2 A at A = 45° using sin 45° = √2/2 and tan 45° = 1 yields the same numerical result, confirming that the simplification sin^2 A + tan^2 A is correct.

Why Other Options Are Wrong:
Option A (2 tan^2 A) and Option B (2 sin^2 A) miscount the factors. Option C (sin^2 A * tan^2 A) multiplies the two terms instead of adding them. Option E (sec^2 A − 1) is equal to tan^2 A alone, which does not include the sin^2 A term we derived. Only Option D matches exactly the simplified expression tan^2 A + sin^2 A.

Common Pitfalls:
One frequent error is to stop at sec^2 A − cos^2 A and try to match it directly to the options without further simplification. Another mistake is mis applying identities, for example, writing sec^2 A as 1 + sin^2 A or confusing tan^2 A with sin^2 A / cos A. Carefully using standard identities and performing algebraic steps one at a time avoids these issues.

Final Answer:
The given product simplifies to sec^2 A − cos^2 A, which further reduces to sin^2 A + tan^2 A.