If (1 + sin A)/cos A + cos A/(1 + sin A) = x in trigonometry, then what is the simplified value of x?

Introduction / Context:
This question tests the ability to simplify a trigonometric expression involving a sum of two rational terms in sine and cosine. Such identities are often required in competitive exams to reduce expressions into a compact form involving standard functions like sec A and tan A.

Given Data / Assumptions:

• Expression: (1 + sin A)/cos A + cos A/(1 + sin A) = x
• A is an angle where the denominators cos A and 1 + sin A are non zero.
• We must simplify x into a standard trigonometric form.

Concept / Approach:
We add the two fractions by taking a common denominator. Then we use the identity 1 - sin^2 A = cos^2 A to simplify. At the end, we rewrite the result using sec A = 1 / cos A. This process reveals a simple final result for x.

Step-by-Step Solution:
Start with x = (1 + sin A)/cos A + cos A/(1 + sin A) Take common denominator cos A(1 + sin A) Numerator becomes (1 + sin A)^2 + cos^2 A Expand (1 + sin A)^2 = 1 + 2 sin A + sin^2 A So numerator is 1 + 2 sin A + sin^2 A + cos^2 A Use identity sin^2 A + cos^2 A = 1 Then numerator becomes 1 + 2 sin A + 1 = 2 + 2 sin A = 2(1 + sin A) Denominator is cos A(1 + sin A) Cancel the factor (1 + sin A) in numerator and denominator Result is x = 2 / cos A = 2 sec A

Verification / Alternative check:
Choose an angle such as A = 30 degrees. Left side: (1 + 1/2)/(√3/2) + (√3/2)/(1 + 1/2) = (3/2)/(√3/2) + (√3/2)/(3/2). This simplifies to (3/2)*(2/√3) + (√3/2)*(2/3) = 3/√3 + √3/3 = √3 + √3/3 = 4√3/3. Right side: 2 sec 30° = 2*(2/√3) = 4/√3 = 4√3/3. Both sides match, confirming the simplification.

Why Other Options Are Wrong:
Options a and b involve cosec A and cosec^2 A, which do not appear naturally in the simplification once we rewrite everything in terms of sine and cosine. Option c, 2 sec^2 A, corresponds to 2(1 + tan^2 A), which is larger than the simplified expression. Option e, 2 tan A, depends on sin A / cos A and does not match the structure of the result.

Common Pitfalls:
Students may miss the cancellation of 1 + sin A or may expand (1 + sin A)^2 incorrectly. Some also forget to apply sin^2 A + cos^2 A = 1, leaving the expression more complicated than necessary. Carefully performing each algebraic step and looking for common factors helps to reach the compact form 2 sec A.

Final Answer:
The simplified value of x is 2 sec A.