Simplify and find the value of the trigonometric expression (sin 7x - sin 5x)/(cos 7x + cos 5x) - (cos 6x - cos 4x)/(sin 6x + sin 4x).

Introduction / Context:
This trigonometry question checks your familiarity with sum and difference formulas and your ability to simplify complex expressions by converting sums and differences of sines and cosines into products. The goal is to reduce the given expression to a simple function of x, ideally involving a single basic trigonometric ratio.

Given Data / Assumptions:

• Expression: (sin 7x - sin 5x)/(cos 7x + cos 5x) - (cos 6x - cos 4x)/(sin 6x + sin 4x)
• x is a real angle and denominators are assumed to be non zero.
• Angles are in radians or degrees consistently; the identities used are valid in both systems.

Concept / Approach:
We use two standard trigonometric identities:
sin C - sin D = 2 cos((C + D)/2) sin((C - D)/2)
cos C + cos D = 2 cos((C + D)/2) cos((C - D)/2)
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)
The idea is to apply these identities to each fraction separately. Many factors will cancel, leaving a simpler result. After simplification, we look for a direct match with basic functions like tan x, 2 tan x or tan 2x.

Step-by-Step Solution:
Step 1: For the first fraction, use C = 7x and D = 5x. Step 2: Apply sin 7x - sin 5x = 2 cos((7x + 5x)/2) sin((7x - 5x)/2) = 2 cos(6x) sin(x). Step 3: Apply cos 7x + cos 5x = 2 cos((7x + 5x)/2) cos((7x - 5x)/2) = 2 cos(6x) cos(x). Step 4: Divide to get (sin 7x - sin 5x)/(cos 7x + cos 5x) = [2 cos(6x) sin(x)]/[2 cos(6x) cos(x)] = tan x. Step 5: For the second fraction, use C = 6x and D = 4x. Step 6: Apply cos 6x - cos 4x = -2 sin((6x + 4x)/2) sin((6x - 4x)/2) = -2 sin(5x) sin(x). Step 7: Apply sin 6x + sin 4x = 2 sin((6x + 4x)/2) cos((6x - 4x)/2) = 2 sin(5x) cos(x). Step 8: Divide to get (cos 6x - cos 4x)/(sin 6x + sin 4x) = [-2 sin(5x) sin(x)]/[2 sin(5x) cos(x)] = -tan x. Step 9: Now combine both parts of the original expression: tan x - ( -tan x ) = tan x + tan x = 2 tan x.

Verification / Alternative check:
You can pick a convenient value of x, such as x = 30 degrees or x = pi/6 radians, and evaluate the original expression numerically using a calculator. The computed value will be very close to 2 tan x for that same x, confirming that the symbolic simplification is correct over the allowed domain where denominators are not zero.

Why Other Options Are Wrong:
1 would require the two fractions to be negatives of each other with magnitude 1/2, which is not the case here. tan 2x and tan(3x/2) involve different angle-doubling or multiple angle relationships and do not match the simplified expression. The value 0 would occur only if the two fractions were equal, but the simplification clearly shows that one is tan x and the other is -tan x, making their difference 2 tan x, not zero.

Common Pitfalls:
A frequent error is to try to expand everything using basic sine and cosine formulas instead of sum to product identities, which leads to messy and error prone algebra. Another issue is forgetting the negative sign in the identity for cos C - cos D. Carefully writing each identity and cancelling common factors step by step ensures an accurate and elegant simplification.

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