Simplify the trigonometric expression 1 / (sec A + tan A). This simplified expression is equal to which of the following?

Introduction / Context:
This question examines your grasp of trigonometric identities and algebraic manipulation. You are asked to simplify 1 / (sec A + tan A) and express it in a simpler trigonometric form. A classic identity shows that (sec A + tan A)(sec A − tan A) = 1, which directly leads to the desired simplification. Recognising and applying difference-of-squares patterns with secant and tangent is the key here.

Given Data / Assumptions:

• The expression is 1 / (sec A + tan A).
• A is an angle for which sec A and tan A are defined and the denominator sec A + tan A is non zero.
• Standard trigonometric identities hold, including sec^2 A − tan^2 A = 1.
• We must match the simplified expression with one of the given options.

Concept / Approach:
We use the identity sec^2 A − tan^2 A = 1, which can be factored as (sec A + tan A)(sec A − tan A) = 1. If this product equals 1, then the reciprocal of sec A + tan A must be sec A − tan A. This logic mirrors the identity involving cosec A and cot A. By focusing on the difference of squares structure, we avoid messy conversions into sine and cosine.

Step-by-Step Solution:
Start from the identity sec^2 A − tan^2 A = 1.Factor the left-hand side as a difference of squares: (sec A + tan A)(sec A − tan A) = 1.Now divide both sides by (sec A + tan A), which is non zero in the domain considered.This gives sec A − tan A = 1 / (sec A + tan A).Therefore 1 / (sec A + tan A) simplifies to sec A − tan A.Among the options, this matches option c.

Verification / Alternative check:
We can verify the result numerically by choosing a simple angle, such as A = 45°. We know cos 45° = sin 45° = √2/2, so sec 45° = √2 and tan 45° = 1. Then sec A + tan A = √2 + 1, and 1 / (sec A + tan A) = 1 / (√2 + 1). Multiply numerator and denominator by (√2 − 1): 1 / (√2 + 1) = (√2 − 1)/(2 − 1) = √2 − 1. Now compute sec A − tan A directly: √2 − 1, which matches the simplified expression, confirming the identity.

Why Other Options Are Wrong:

• cosecA − cotA is analogous to the reciprocal of cosec A + cot A, not sec A + tan A.
• sinA − cosA and sinA + cosA are unrelated linear combinations of sine and cosine and do not match this identity.
• cosecA + cotA is not equal to 1 / (secA + tanA); it belongs to a different set of identities.

Common Pitfalls:

• Trying to rewrite everything in terms of sin and cos unnecessarily, which can lead to algebraic errors.
• Confusing the identity sec^2 A − tan^2 A = 1 with tan^2 A + 1 = sec^2 A and misusing it.
• Assuming the answer must contain sine or cosine explicitly instead of recognising a compact form in sec A − tan A.

Final Answer:
secA - tanA