Simplify the trigonometric expression: (sec(A) - 1) / (sec(A) + 1) and choose an equivalent expression in terms of sin(A) or cos(A).

Introduction / Context:
This trigonometric simplification question asks you to rewrite the expression (sec(A) − 1) / (sec(A) + 1) in terms of sine or cosine. Simplifying such expressions using algebraic manipulation and fundamental identities is a common task in trigonometry and aptitude exams. The goal is to recognize how to convert secant into cosine and then simplify the resulting fraction to match one of the given equivalent forms.

Given Data / Assumptions:

• Expression: (sec(A) − 1) / (sec(A) + 1).
• A is an angle for which sec(A) is defined and sec(A) + 1 ≠ 0.
• We must express this fraction in terms of sin(A) or cos(A) and match it to one of the options.

Concept / Approach:
We start by expressing sec(A) in terms of cos(A) using sec(A) = 1 / cos(A). Then we simplify the resulting expression. A useful technique is to multiply the numerator and denominator by (sec(A) − 1), creating a difference of squares in the denominator. After simplification, we convert the result into a form involving cosine and possibly sine by using the identity sin^2(A) + cos^2(A) = 1. Finally, we compare the simplified form with the options to identify the correct match.

Step-by-Step Solution:
1) Let E = (sec(A) − 1) / (sec(A) + 1). 2) Multiply numerator and denominator by (sec(A) − 1) to simplify: E = (sec(A) − 1)^2 / (sec^2(A) − 1). 3) Use the identity sec^2(A) − 1 = tan^2(A). 4) So E = (sec(A) − 1)^2 / tan^2(A). 5) Now write sec(A) and tan(A) in terms of sin(A) and cos(A): sec(A) = 1 / cos(A) and tan(A) = sin(A) / cos(A). 6) Then (sec(A) − 1) = (1 / cos(A)) − 1 = (1 − cos(A)) / cos(A). 7) Thus (sec(A) − 1)^2 = (1 − cos(A))^2 / cos^2(A). 8) Also tan^2(A) = sin^2(A) / cos^2(A). 9) Substitute into E: E = [(1 − cos(A))^2 / cos^2(A)] / [sin^2(A) / cos^2(A)] = (1 − cos(A))^2 / sin^2(A). 10) Use sin^2(A) = 1 − cos^2(A) = (1 − cos(A))(1 + cos(A)) to get: E = (1 − cos(A))^2 / [(1 − cos(A))(1 + cos(A))] = (1 − cos(A)) / (1 + cos(A)).

Verification / Alternative check:
Pick a convenient angle, for example A = 60°. Then sec(60°) = 2, so the original expression (sec(A) − 1) / (sec(A) + 1) becomes (2 − 1) / (2 + 1) = 1 / 3. Now evaluate (1 − cos(A)) / (1 + cos(A)) at A = 60°. Here cos(60°) = 1/2, so (1 − 1/2) / (1 + 1/2) = (1/2) / (3/2) = 1/3, which matches the original value. This numerical check supports the algebraic simplification.

Why Other Options Are Wrong:
Option a, (1 − sin(A)) / (1 + sin(A)), gives a different value when tested with A = 60°, as does option c, (1 + sin(A)) / (1 − sin(A)). Option b, (1 + cos(A)) / (1 − cos(A)), is the reciprocal of the correct expression. Option e, sin(A) / cos(A), is simply tan(A) and does not match the simplified result. Only option d, (1 − cos(A)) / (1 + cos(A)), matches the derived form exactly.

Common Pitfalls:
Students sometimes try to simplify directly without multiplying by the conjugate, which can make the algebra messy. Another common mistake is to cancel terms incorrectly instead of factoring and canceling common factors. Misusing the identity sin^2(A) + cos^2(A) = 1 is also frequent. Following a structured approach—rewrite in terms of cos(A), use the difference of squares, and apply the Pythagorean identity—helps avoid these errors.

Final Answer:
The expression (sec(A) − 1) / (sec(A) + 1) simplifies to (1 - cos(A)) / (1 + cos(A)).