If 1 + sec A = x for an acute angle A, express x completely in terms of sin A, tan A and cos A using trigonometric identities.

Introduction / Context:
This question is about rewriting a simple trigonometric expression involving the secant function in terms of sine, cosine, and tangent. It tests your ability to use basic identities and algebraic manipulation to transform one expression into another equivalent form. Such rewriting is frequently used to simplify complex expressions or to match forms given in multiple choice questions.

Given Data / Assumptions:
- We are told that 1 + sec A = x for an acute angle A.
- We must rewrite this quantity using sin A, tan A, and cos A, then compare with the given options.
- The angle A is acute, so sin A and cos A are positive and all expressions are well defined.

Concept / Approach:
Secant is defined as sec A = 1 / cos A. We want to use algebra to rewrite 1 + sec A in a form involving sin A and tan A. Because tan A = sin A / cos A and 1 − cos A is related to sin squared through the identity 1 − cos A = 2 sin²(A/2), an approach is to set up a rational expression and use the identity for 1 − cos A expressed in terms of sin A and cos A.

Step-by-Step Solution:
Step 1: Start with the expression 1 + sec A. Step 2: Replace sec A by 1 / cos A to get 1 + 1 / cos A. Step 3: Combine the terms over a common denominator cos A: 1 + 1 / cos A = (cos A + 1) / cos A. Step 4: We want an expression involving sin A and tan A. Note that tan A = sin A / cos A and sin² A = 1 − cos² A, but here it is more helpful to manipulate the expression sin A tan A / (1 − cos A). Step 5: Consider sin A tan A / (1 − cos A). Substitute tan A = sin A / cos A to obtain (sin A * sin A / cos A) / (1 − cos A) = (sin² A / cos A) / (1 − cos A). Step 6: Recognise that 1 − cos A can be written as sin² A / (1 + cos A). Substituting this into the denominator gives (sin² A / cos A) / (sin² A / (1 + cos A)) = (sin² A / cos A) * ((1 + cos A) / sin² A). Step 7: Cancel sin² A from numerator and denominator to obtain (1 + cos A) / cos A, which is exactly 1 + sec A. Step 8: Therefore 1 + sec A is equal to sin A tan A / (1 − cos A), so x = sin A tan A / (1 − cos A).
Verification / Alternative check:
You can test this identity with a specific value, for example A = 60 degrees. Then 1 + sec 60° = 1 + 2 = 3. On the other hand sin 60° = √3 / 2, tan 60° = √3, and cos 60° = 1/2. Plugging into sin A tan A / (1 − cos A) gives (√3 / 2 * √3) / (1 − 1/2) = (3 / 2) / (1/2) = 3, confirming the identity numerically.

Why Other Options Are Wrong:
Option a uses 1 + cos A instead of 1 − cos A in the denominator, which produces (1 − cos A) / cos A, a different expression.
Options b and d introduce square roots that do not appear in the simplified result.
Option e, cos A / (1 − cos A), has a different structure and does not equal 1 + sec A for general A.

Common Pitfalls:
Students often make mistakes when replacing sec A with 1 / cos A, forgetting to apply the common denominator correctly or misusing the identity for 1 − cos A. Another frequent error is to cancel terms incorrectly in complex fractions. Writing each algebraic step carefully and checking with a sample angle like 60 degrees helps to avoid such errors and reinforces understanding of the identities.

Final Answer:
The correct equivalent expression is x = sin A tan A / (1 − cos A).