If 1 / (1 + cos A) + 1 / (1 − cos A) = x, then what is the simplified expression for x in terms of A?

Introduction / Context:
This problem focuses on simplifying an expression involving cos A in the denominators and writing the result in terms of a standard trigonometric function. It uses identities connecting cosine and sine and shows how expressions with reciprocal denominators can be combined neatly.

Given Data / Assumptions:

• x = 1 / (1 + cos A) + 1 / (1 − cos A).
• Angle A is such that denominators are non zero.
• We must write x in terms of a single trigonometric function.

Concept / Approach:
Main ideas:

• Use a common denominator (1 + cos A)(1 − cos A).
• Recognise that (1 + cos A)(1 − cos A) = 1 − cos^2 A = sin^2 A.
• Simplify the numerator after combining fractions.
• Express the result in terms of cosec A using the identity cosec^2 A = 1 / sin^2 A.

Step-by-Step Solution:
Start with x = 1 / (1 + cos A) + 1 / (1 − cos A). Common denominator is (1 + cos A)(1 − cos A) = 1 − cos^2 A = sin^2 A. Rewrite the numerators: x = (1 − cos A + 1 + cos A) / sin^2 A. Add the terms in the numerator: 1 − cos A + 1 + cos A = 2. So x = 2 / sin^2 A. Recall that cosec^2 A = 1 / sin^2 A. Therefore x = 2 * (1 / sin^2 A) = 2cosec^2 A.

Verification / Alternative check:
Take a test angle, for example A = 30°. Then cos 30° = √3 / 2 and sin 30° = 1 / 2. Evaluate x numerically: 1 / (1 + √3 / 2) + 1 / (1 − √3 / 2). This simplifies to a numerical value equal to 2 / sin^2 30°, which is 2 / (1 / 4) = 8. Meanwhile, 2cosec^2 30° = 2 * (1 / (1 / 4)) = 8. The two values match, confirming the identity.

Why Other Options Are Wrong:
Option a: 2sec^2 A uses cos A in the denominator but does not match 2 / sin^2 A.
Option b: 2cosec A would equal 2 / sin A, missing the square on sine.
Option d and option e: 2sec A or 4sec A involve cosine in the denominator directly and do not correspond to 2 / sin^2 A.

Common Pitfalls:
Many students forget that (1 + cos A)(1 − cos A) equals sin^2 A and instead try to expand incorrectly. Another common issue is mishandling the numerator and failing to combine the terms properly. Keeping track of standard identities and performing algebra carefully helps avoid such problems.

Final Answer:
The simplified expression for x is 2cosec^2 A.