Evaluate the trigonometric expression using exact values of standard angles: (cos 40° - cos 140°) / (sin 80° + sin 20°). Choose the correct simplified value.

Introduction / Context:
This question asks you to simplify a trigonometric expression involving cos 40°, cos 140°, sin 80° and sin 20°. It is designed to test your understanding of sum to product identities and your familiarity with exact values of trigonometric functions for special angles. Rather than plugging approximate decimal values, the goal is to manipulate the expression symbolically so that it reduces to a simple surd form such as 2 divided by the square root of 3.

Given Data / Assumptions:

• Expression: E = (cos 40° - cos 140°) / (sin 80° + sin 20°).
• All angles are measured in degrees.
• Standard trigonometric identities for sums and differences are applicable.
• Exact trigonometric values for 30°, 60° and other related angles are known.

Concept / Approach:
We use the following sum to product identities: cos C - cos D = -2 sin((C + D) / 2) sin((C - D) / 2), sin C + sin D = 2 sin((C + D) / 2) cos((C - D) / 2). By applying these identities to the numerator and denominator separately, common factors will appear that can be cancelled. After this simplification, we will be left with a ratio involving standard angles, which can be evaluated using known exact values for sine and cosine at 30° and 60°.

Step-by-Step Solution:
Step 1: Simplify the numerator using cos C - cos D: cos 40° - cos 140° = -2 sin((40° + 140°) / 2) sin((40° - 140°) / 2). Step 2: Compute the averages and differences: (40° + 140°) / 2 = 90°, (40° - 140°) / 2 = -50°. Step 3: So the numerator N = -2 sin 90° * sin(-50°) = -2 * 1 * (-sin 50°) = 2 sin 50°. Step 4: Simplify the denominator using sin C + sin D: sin 80° + sin 20° = 2 sin((80° + 20°) / 2) cos((80° - 20°) / 2). Step 5: Compute the averages: (80° + 20°) / 2 = 50°, (80° - 20°) / 2 = 30°. Step 6: So the denominator D = 2 sin 50° cos 30°. Step 7: Form the ratio: E = N / D = [2 sin 50°] / [2 sin 50° cos 30°] = 1 / cos 30°. Step 8: Use the exact value cos 30° = √3 / 2, so 1 / cos 30° = 1 / (√3 / 2) = 2 / √3.

Verification / Alternative check:
Using a scientific calculator, you can compute cos 40°, cos 140°, sin 80° and sin 20° in degree mode and substitute into the original expression. The numerical value obtained will be approximately 1.1547. Next, compute 2 / √3, which also evaluates to about 1.1547. Since the symbolic manipulation is correct and the numerical check agrees, the simplified expression is indeed 2 / √3.

Why Other Options Are Wrong:
Option a, 2√3, would be much larger (about 3.464) and does not match the numerical value of the expression. Option c, 1 / √3, gives approximately 0.577, which is too small. Option d, √3, is about 1.732, larger than the actual result. Option e, 1/2, clearly does not match. Only option b, 2/√3, fits both the algebraic derivation and the numerical approximation.

Common Pitfalls:
One trap is to attempt to evaluate all trigonometric values in decimal form too early, which can obscure the underlying structure and lead to rounding errors. Another common mistake is incorrect use of the sum to product formulas, especially mixing up sine and cosine versions or miscalculating the averaged angles. Always write the identities clearly before substitution, and pay attention to the signs of angles, for example sin(-50°) = -sin 50°. Careful handling of these details leads quickly to a clean exact form.

Final Answer:
Therefore, the value of the expression is 2/√3.