Simplify the trigonometric expression: (sin θ / (1 + cos θ)) + (sin θ / (1 - cos θ)). Find its exact value in simplest form (assume denominators are non-zero).

Introduction / Context: This question tests algebraic simplification of trigonometric fractions. The expression has two terms with similar structure, and the most efficient method is to combine them over a common denominator. The key identity that unlocks the simplification is 1 - cos^2 θ = sin^2 θ. Once you combine and reduce, the result becomes a simple reciprocal trig function. These problems are common in aptitude because they reward identity recognition and careful cancellation.

Given Data / Assumptions:

• Expression: (sin θ / (1 + cos θ)) + (sin θ / (1 - cos θ)) • Assume 1 + cos θ ≠ 0 and 1 - cos θ ≠ 0 • Required: simplified exact value

Concept / Approach: Take sin θ as a common factor and add the remaining fractions: sin θ * (1/(1 + cos θ) + 1/(1 - cos θ)). Combine using the common denominator (1 + cos θ)(1 - cos θ) = 1 - cos^2 θ. Then replace 1 - cos^2 θ with sin^2 θ and cancel one sin θ factor, giving 2/sin θ = 2 cosec θ.

Step-by-Step Solution: 1) Start: sin θ/(1 + cos θ) + sin θ/(1 - cos θ) 2) Factor out sin θ: = sin θ * [1/(1 + cos θ) + 1/(1 - cos θ)] 3) Combine the bracketed fractions: 1/(1 + cos θ) + 1/(1 - cos θ) = [(1 - cos θ) + (1 + cos θ)] / [(1 + cos θ)(1 - cos θ)] 4) Simplify numerator: (1 - cos θ) + (1 + cos θ) = 2 5) Simplify denominator using difference of squares: (1 + cos θ)(1 - cos θ) = 1 - cos^2 θ 6) So the bracket becomes: 2 / (1 - cos^2 θ) 7) Use identity 1 - cos^2 θ = sin^2 θ: = 2 / sin^2 θ 8) Multiply by sin θ: sin θ * (2/sin^2 θ) = 2/sin θ = 2 cosec θ

Verification / Alternative check: Take θ = 30°: sin 30° = 1/2, cos 30° = √3/2. LHS = (1/2)/(1 + √3/2) + (1/2)/(1 - √3/2). Both terms simplify to a value whose sum equals 4. RHS = 2 cosec 30° = 2 * 2 = 4. The values match, confirming the simplification.

Why Other Options Are Wrong: • 2 sec θ would require the expression to reduce to 2/cos θ, but the identity produces 2/sin θ. • 2 sin θ and 2 cos θ are not reciprocal forms and do not match after combining denominators. • 0 is impossible unless the two terms cancel exactly, which they do not.

Common Pitfalls: • Forgetting that (1 + cos θ)(1 - cos θ) is a difference of squares. • Replacing 1 - cos^2 θ incorrectly (it equals sin^2 θ, not sin θ). • Cancelling sin θ incorrectly across addition instead of after combining into a single fraction.

Final Answer: 2 cosec θ