In trigonometry, if (1 + cos A)/(1 - cos A) = x, then which of the following expressions correctly represents the value of x?

Introduction / Context:
This question checks understanding of trigonometric identities and algebraic manipulation of expressions involving cosine, tangent, and secant. The expression (1 + cos A)/(1 - cos A) often appears in simplification problems, and candidates need to convert it into a form involving standard functions like tan A and sec A.

Given Data / Assumptions:

• (1 + cos A)/(1 - cos A) = x
• A is an angle in a domain where all expressions involved are defined

Concept / Approach:
We use the identity 1 - cos A = 2 sin^2(A/2) and 1 + cos A = 2 cos^2(A/2). Their ratio gives a half angle expression. We then rewrite the result in terms of tan A and sec A using the identities sec A = 1 / cos A and tan A = sin A / cos A, along with 1 + tan^2 A = sec^2 A.

Step-by-Step Solution:
(1 + cos A) = 2 cos^2(A/2) (1 - cos A) = 2 sin^2(A/2) So x = (1 + cos A)/(1 - cos A) = [2 cos^2(A/2)] / [2 sin^2(A/2)] This simplifies to x = cot^2(A/2) Now express cot^2(A/2) in terms of tan A and sec A Use the relation: cot(A/2) = (1 - cos A) / sin A Also express tan A and sec A in basic sine and cosine and manipulate to match the option tan^2 A / (sec A - 1)^2 After simplification one finds that cot^2(A/2) = tan^2 A / (sec A - 1)^2

Verification / Alternative check:
We can verify by choosing a specific angle, for example A = 60 degrees. Compute the left side (1 + cos 60)/(1 - cos 60) and the candidate right side tan^2 60 / (sec 60 - 1)^2. Both evaluate to the same numerical result, which confirms the identity for at least one nontrivial angle and supports the algebraic derivation.

Why Other Options Are Wrong:
Option a gives a form with cot^2 A instead of tan^2 A and does not match the derived relation. Option b and option c have the wrong sign or denominator and fail when tested with a concrete angle such as 60 degrees. Option e introduces sec^2 A in the denominator and does not simplify to the required ratio for general A.

Common Pitfalls:
A common mistake is to confuse half angle identities with double angle identities or to forget that 1 + cos A and 1 - cos A can be expressed in terms of sin(A/2) and cos(A/2). Another frequent error is incorrect squaring of expressions like (sec A - 1), which changes signs and leads to an incorrect denominator. Students may also mix up tan^2 A and cot^2 A because of their reciprocal relationship.

Final Answer:
Therefore, the correct expression for x is tan^2 A / (sec A - 1)^2.