In trigonometric simplification, evaluate the expression {(sin 4x + sin 4y) * tan(2x - 2y)} / (sin 4x - sin 4y). Which of the following options gives the correct simplified form of this expression in terms of x and y?

Introduction / Context:
This question tests standard trigonometric identities, especially sum to product formulas and the relationship between tangent, sine and cosine. The expression involves sin 4x, sin 4y and tan(2x - 2y), and the goal is to simplify it to a much more compact form in terms of x and y. Such simplification problems are very common in aptitude exams and help reinforce fluency with classical trigonometric identities.

Given Data / Assumptions:

• The expression is E = {(sin 4x + sin 4y) * tan(2x - 2y)} / (sin 4x - sin 4y).
• Angles are assumed to be in the same unit (degrees or radians) consistently.
• sin 4x - sin 4y is assumed non zero so that the division is defined.
• Standard trigonometric identities for sine and tangent apply.

Concept / Approach:
We use two main ideas. First, we express sums and differences of sines using sum to product identities: sin C + sin D = 2 sin((C + D) / 2) cos((C - D) / 2) sin C - sin D = 2 cos((C + D) / 2) sin((C - D) / 2) Second, we write tangent in terms of sine and cosine: tan T = sin T / cos T By substituting these into the expression, many factors cancel and we reach a simple tangent of a single angle.

Step-by-Step Solution:
Step 1: Apply the sum formula to sin 4x + sin 4y: sin 4x + sin 4y = 2 sin(2x + 2y) cos(2x - 2y). Step 2: Apply the difference formula to sin 4x - sin 4y: sin 4x - sin 4y = 2 cos(2x + 2y) sin(2x - 2y). Step 3: Replace tan(2x - 2y) by sin(2x - 2y) / cos(2x - 2y). Step 4: Substitute into E: E = [2 sin(2x + 2y) cos(2x - 2y) * (sin(2x - 2y) / cos(2x - 2y))] / [2 cos(2x + 2y) sin(2x - 2y)]. Step 5: Cancel common factors: the 2 in numerator and denominator cancel, cos(2x - 2y) cancels, and sin(2x - 2y) cancels. Step 6: The remaining expression is: E = sin(2x + 2y) / cos(2x + 2y) = tan(2x + 2y).

Verification / Alternative check:
We can pick convenient values for x and y, for example x = 30° and y = 10°, evaluate the original expression with a calculator, and then evaluate tan(2x + 2y) = tan(80°). Both results match numerically. Since the trigonometric identities used are exact and the simplification is purely algebraic, equality will hold for all allowed values of x and y, not just for particular numerical choices.

Why Other Options Are Wrong:
Option b, tan(2x - 2y), would appear if we did not use the correct combination of sum and difference formulas or if we mistakenly cancelled wrong factors. Option c, cot(x - y), and option e, cot(2x + 2y), use cotangent and are not consistent with the final ratio sin(2x + 2y) / cos(2x + 2y). Option d, tan(4x), ignores the presence of y entirely and does not reflect the symmetric dependence on both x and y in the original expression. Only tan(2x + 2y) correctly emerges from the systematic simplification.

Common Pitfalls:
The most common error is to misapply the sum to product formulas, especially swapping sine and cosine in the expressions for sin C ± sin D. Another frequent mistake is to forget to convert tangent into sine and cosine, which obscures the common factors that must cancel. Students sometimes cancel terms that are added instead of multiplied, which is algebraically invalid. Writing every intermediate step clearly and keeping track of which parts multiply and which parts add or subtract helps avoid these errors.

Final Answer:
Therefore, the simplified value of the expression is tan(2x + 2y).