Simplify {[sin (x + y) – 2 sin x + sin (x – y)] / [cos (x – y) + cos (x + y) – 2 cos x]} × [(sin 10x – sin 8x) / (cos 10x + cos 8x)].

Introduction / Context:
This problem combines several trigonometric identities. The expression contains sin and cos of sums and differences, as well as sin and cos of multiple angles. The goal is to simplify the entire product to a compact form. This tests familiarity with sum to product identities and with the basic relationship between sine, cosine, and tangent.

Given Data / Assumptions:

• Expression E = {[sin(x + y) - 2 sin x + sin(x - y)] / [cos(x - y) + cos(x + y) - 2 cos x]} × [(sin 10x - sin 8x) / (cos 10x + cos 8x)].
• x and y are real angles for which denominators are not zero.
• We need to simplify E.

Concept / Approach:
We use the identity sin(A + B) + sin(A - B) = 2 sin A cos B to simplify the sine terms in the numerator of the first fraction, and cos(A + B) + cos(A - B) = 2 cos A cos B for the denominator. After simplifying, the first fraction will reduce to tan x. For the second fraction, we use sin C - sin D and cos C + cos D formulas, which also simplify to tan x. The product then becomes tan x × tan x = tan^2 x.

Step-by-Step Solution:
Consider sin(x + y) + sin(x - y). Using sin(A + B) + sin(A - B) = 2 sin A cos B with A = x and B = y, we get sin(x + y) + sin(x - y) = 2 sin x cos y. So the numerator of the first fraction is 2 sin x cos y - 2 sin x = 2 sin x (cos y - 1). In the denominator, cos(x - y) + cos(x + y) = 2 cos x cos y by the identity cos(A + B) + cos(A - B) = 2 cos A cos B. Thus the denominator becomes 2 cos x cos y - 2 cos x = 2 cos x (cos y - 1). The first fraction simplifies to [2 sin x (cos y - 1)] / [2 cos x (cos y - 1)] = tan x. Now simplify the second fraction. Use sin C - sin D = 2 cos((C + D)/2) sin((C - D)/2) with C = 10x and D = 8x. Then sin 10x - sin 8x = 2 cos(9x) sin(x). Use cos C + cos D = 2 cos((C + D)/2) cos((C - D)/2) for the denominator: cos 10x + cos 8x = 2 cos(9x) cos(x). The second fraction becomes [2 cos(9x) sin x] / [2 cos(9x) cos x] = tan x. Therefore the entire expression E = tan x × tan x = tan^2 x.

Verification / Alternative check:
Choose a convenient angle, for example x = 30 degrees and y = 20 degrees, and evaluate both the original expression and tan^2 x numerically using a calculator. The computed values match, which confirms the simplification.

Why Other Options Are Wrong:
2 tan x: This would be the result if one of the fractions simplified to 1 and the other to tan x, but both fractions simplify to tan x, not to 1.
1: This would require the two fractions to be reciprocals of each other, which is not the case.
0: The expression is not identically zero for general x and y, since both numerators are non zero for typical angles.
sec^2 x: Although sec^2 x relates to tan^2 x through sec^2 x = 1 + tan^2 x, the exact simplification here is just tan^2 x and not tan^2 x plus 1.

Common Pitfalls:
A common error is mixing up the sum to product formulas for sine and cosine, or applying them with the wrong signs. Another mistake is failing to factor out the full common term in both numerator and denominator, leading to incomplete cancellation. Being systematic with the identities and double checking each step helps avoid these mistakes.

Final Answer:
The simplified value of the given expression is tan^2 x.