Simplify and evaluate the trigonometric expression: “What is the value of [sec θ / (sec θ − 1)] + [sec θ / (sec θ + 1)] ?”

Introduction / Context:
This question tests algebraic manipulation of trigonometric expressions, a frequent topic in quantitative aptitude and mathematics sections of competitive exams. The expression involves the secant function and asks for simplification to a more familiar trigonometric form. To solve it, we must use algebraic techniques like combining fractions and basic trigonometric identities relating sec θ, tan θ, and cosec θ.

Given Data / Assumptions:

• We need to simplify the expression: [sec θ / (sec θ − 1)] + [sec θ / (sec θ + 1)].
• θ is an angle for which the trigonometric functions involved are defined, and sec θ is not equal to ±1, so that denominators are non zero.
• Four possible simplified forms are provided as options.
• We will use standard trigonometric identities: sec^2θ = 1 + tan^2θ and 1 + cot^2θ = cosec^2θ.

Concept / Approach:
The key steps are to combine the two fractions over a common denominator and then express everything in terms of sec^2θ. After that, we link sec^2θ with tan^2θ using the identity sec^2θ = 1 + tan^2θ. Finally, we transform the resulting expression into terms of cot^2θ and cosec^2θ to match one of the given options. Careful algebraic manipulation reveals that the entire expression simplifies neatly to 2 cosec^2θ.

Step-by-Step Solution:

Verification / Alternative check:
We can test the simplification with a specific angle where all functions are defined. For example, take θ = 45 degrees. Then sec 45° = √2 and cosec 45° = √2. Compute the left side: sec θ / (sec θ − 1) + sec θ / (sec θ + 1) = √2 / (√2 − 1) + √2 / (√2 + 1). Multiply numerator and denominator appropriately or use rationalisation, and we find that the value simplifies to 4. Now compute the right side for the option 2 cosec^2θ: cosec 45° = √2, so cosec^2θ = 2, and 2 * 2 = 4. The numerical values match, giving strong confirmation that the algebraic simplification is correct.

Why Other Options Are Wrong:

• Option A: 2 sin2 θ is not equivalent to the derived expression. For θ = 45 degrees, sin 2θ is sin 90 degrees, which is 1, so 2 sin2 θ gives 2, not 4.
• Option B: 2(1 + tan2θ) is a different function and does not match the simplified result. Checking with θ = 45 degrees gives 2(1 + tan 90 degrees), which is undefined.
• Option D: sin2θ is again much smaller than 2 cosec^2θ for most valid angles. For θ = 45 degrees it equals 1, not 4.

Common Pitfalls:
Candidates may make algebraic errors when combining fractions, such as adding denominators directly or forgetting to simplify the numerator properly. Another common mistake is to stop after obtaining 2 sec^2θ / (sec^2θ − 1) and not converting it further into a simpler standard form. It is also easy to confuse tan^2θ and cot^2θ identities. A systematic approach is to always look for basic identities like sec^2θ = 1 + tan^2θ and 1 + cot^2θ = cosec^2θ and to check the final answer by substituting a simple angle where all functions are defined. This ensures both algebraic and conceptual correctness.

Final Answer:
The expression [sec θ / (sec θ − 1)] + [sec θ / (sec θ + 1)] simplifies to 2 cosec^2 θ.