If 2 sec A − (1 + sin A) / cos A = x, simplify this trigonometric expression and find x in its simplest form.

Introduction / Context:
This problem involves simplifying a trigonometric expression that contains secant, sine, and cosine. It tests your ability to rewrite reciprocal functions in terms of sine and cosine and then use algebraic manipulation to express the result in a compact and recognisable form. Such transformations occur frequently in trigonometry questions that aim to check conceptual understanding rather than mere memorisation.

Given Data / Assumptions:
- The expression is x = 2 sec A − (1 + sin A) / cos A.
- A is an angle for which the trigonometric functions and denominators are defined, typically avoiding cos A = 0.
- We must simplify this expression to one of the forms given in the options.

Concept / Approach:
Secant is the reciprocal of cosine: sec A = 1 / cos A. So 2 sec A can be written as 2 / cos A. The second term already has denominator cos A. Thus, placing both terms over the common denominator cos A allows us to combine them. After simplification, we look for an opportunity to factor or rationalise, aiming for a form such as cos A / (1 + sin A) that matches one of the choices.

Step-by-Step Solution:
Step 1: Rewrite sec A in terms of cosine: sec A = 1 / cos A, so 2 sec A = 2 / cos A. Step 2: The expression becomes x = 2 / cos A − (1 + sin A) / cos A. Step 3: Since both terms share the same denominator cos A, combine them: x = [2 − (1 + sin A)] / cos A. Step 4: Simplify the numerator: 2 − (1 + sin A) = 2 − 1 − sin A = 1 − sin A. Step 5: So x = (1 − sin A) / cos A. Step 6: Recognise that (1 − sin A) / cos A can be further rewritten by multiplying numerator and denominator by (1 + sin A) to obtain a more standard form. Step 7: Multiply numerator and denominator by (1 + sin A): x = [(1 − sin A)(1 + sin A)] / [cos A(1 + sin A)]. Step 8: The numerator becomes 1 − sin² A, and since 1 − sin² A = cos² A, we get x = cos² A / [cos A(1 + sin A)] = cos A / (1 + sin A).
Verification / Alternative check:
To check, choose a convenient angle, for example A = 30 degrees. Compute the original expression: sec 30° = 2 / √3, so 2 sec 30° = 4 / √3. Also sin 30° = 1/2 and cos 30° = √3 / 2, so (1 + sin 30°) / cos 30° = (1 + 1/2) / (√3 / 2) = (3/2) / (√3 / 2) = 3 / √3 = √3. Thus x = 4 / √3 − √3. On the other hand, the simplified form cos A / (1 + sin A) at A = 30 degrees is (√3 / 2) / (1 + 1/2) = (√3 / 2) / (3/2) = √3 / 3. When both original and simplified forms are evaluated numerically and rationalised, they match, confirming the algebraic steps are correct.

Why Other Options Are Wrong:
Option a, cosec A / (1 + sin A), introduces the reciprocal of sine and cannot be obtained from the combined form (1 − sin A) / cos A without creating additional factors that do not cancel properly.
Option c, cos A (1 + sin A), is the product rather than a ratio, so it disagrees dimensionally with the original fraction structure.
Option d, cosec A (1 + sin A), again uses the reciprocal of sine and produces a completely different behaviour as A varies.
Option e, sin A / (1 − cos A), bears no close algebraic relationship to the original expression when simplified.

Common Pitfalls:
The main pitfalls are mishandling the negative sign when combining fractions and forgetting to distribute it across the entire bracket (1 + sin A). Another frequent error is not recognising the Pythagorean identity 1 − sin² A = cos² A when rationalising. Some students also stop at (1 − sin A) / cos A and do not see that an additional step can produce a cleaner expression that matches the options. Practising rationalisation techniques helps in spotting these simplifications quickly.

Final Answer:
The simplified expression is x = cos A / (1 + sin A).