Without assuming specific values of θ and φ, simplify the trigonometric expression (sin θ + sin φ)/(cos θ + cos φ) + (cos θ − cos φ)/(sin θ − sin φ) to its exact value.

Introduction / Context:
This question tests your understanding of trigonometric identities, especially the sum-to-product and difference-to-product formulas for sine and cosine. The goal is to simplify a seemingly complicated expression involving sums and differences of angles without substituting any particular numerical values for θ and φ.

Given Data / Assumptions:

• Expression: (sin θ + sin φ)/(cos θ + cos φ) + (cos θ − cos φ)/(sin θ − sin φ).
• θ and φ are angles for which the denominators are non zero.
• Standard trigonometric identities can be applied (sum and difference formulas).
• We seek a simplified exact value, not a numerical approximation.

Concept / Approach:
We use the sum-to-product identities: sin θ + sin φ = 2 sin((θ + φ)/2) cos((θ − φ)/2), cos θ + cos φ = 2 cos((θ + φ)/2) cos((θ − φ)/2), cos θ − cos φ = −2 sin((θ + φ)/2) sin((θ − φ)/2), sin θ − sin φ = 2 cos((θ + φ)/2) sin((θ − φ)/2). By substituting these into the two fractions, most factors cancel, leaving expressions in tan((θ + φ)/2). The two terms then combine in a simple way.

Step-by-Step Solution:
Let A = (θ + φ)/2 and B = (θ − φ)/2 for convenience. Then sin θ + sin φ = 2 sin A cos B. cos θ + cos φ = 2 cos A cos B. So the first fraction becomes (2 sin A cos B)/(2 cos A cos B) = tan A. Next, cos θ − cos φ = −2 sin A sin B. And sin θ − sin φ = 2 cos A sin B. So the second fraction becomes (−2 sin A sin B)/(2 cos A sin B) = −tan A. Now add the two results: tan A + (−tan A) = 0. Therefore the entire expression simplifies to 0.

Verification / Alternative check:
You can pick specific angles to confirm the result, provided denominators are non zero. For example, take θ = 60° and φ = 30°. Direct substitution into the original expression using known values of sine and cosine yields a numerical result very close to zero (up to rounding), which supports the algebraic simplification. However, the identity based derivation is exact and holds for all valid θ and φ.

Why Other Options Are Wrong:
The value 1 would require the two fractions to combine constructively rather than cancel. A value like 2 or 1/2 would suggest a fixed non zero multiple of some basic ratio, which is not supported once the identities are applied. The option −1 would require tan A + (−tan A) to equal −1, which is impossible as they cancel exactly. Only 0 matches the simplification.

Common Pitfalls:
Learners might try to expand sine and cosine of sums directly, which is messy and error prone, instead of using sum-to-product formulas. Another pitfall is incorrect signs in formulas like cos θ − cos φ, which has a leading negative sign in its identity. Cancelling common factors incorrectly can also lead to wrong non zero answers. Carefully applying the standard identities avoids these issues.

Final Answer:
Thus, the expression simplifies to the exact value 0.