Evaluate the trigonometric expression (sin 59° cos 31° + cos 59° sin 31°)/(cos 20° cos 25° - sin 20° sin 25°).

Introduction / Context:
This trigonometry question is designed to test your ability to use the compound angle formulas for sine and cosine. The expression is written in a way that suggests the direct use of identities for sin(A + B) and cos(A + B), which can greatly simplify what initially looks like a complicated fraction.

Given Data / Assumptions:

• Expression: (sin 59° cos 31° + cos 59° sin 31°)/(cos 20° cos 25° - sin 20° sin 25°)
• Angles are in degrees.
• All trigonometric functions are defined for the angles involved.

Concept / Approach:
We recall the standard compound angle identities:
sin(A + B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B - sin A sin B
The numerators and denominators in the expression match these patterns exactly. So we can convert each part to a single sine or cosine value, and then simplify the resulting fraction to obtain a simple surd value.

Step-by-Step Solution:
Step 1: Identify A and B for the numerator. Take A = 59° and B = 31°. Step 2: Use sin(A + B) = sin A cos B + cos A sin B to rewrite the numerator. Step 3: Compute A + B for the numerator: 59° + 31° = 90°, so the numerator becomes sin 90°. Step 4: Since sin 90° = 1, the entire numerator equals 1. Step 5: For the denominator, take A = 20° and B = 25°. Step 6: Use cos(A + B) = cos A cos B - sin A sin B to rewrite the denominator. Step 7: Compute A + B for the denominator: 20° + 25° = 45°, so the denominator becomes cos 45°. Step 8: Since cos 45° = 1/sqrt(2), the denominator equals 1/sqrt(2). Step 9: The full expression is 1 divided by 1/sqrt(2), which equals sqrt(2).

Verification / Alternative check:
You can check this by using approximate decimal values: sin 59° cos 31° + cos 59° sin 31° approximates to 1, and cos 20° cos 25° - sin 20° sin 25° approximates to 0.707. Dividing 1 by 0.707 gives roughly 1.414, which matches the common decimal value for sqrt(2). This numeric consistency supports the exact symbolic answer.

Why Other Options Are Wrong:
1/sqrt(2) would be the correct answer if the numerator had been 1/2 instead of 1, which is not the case. 2sqrt(2) is simply twice the correct value and does not follow from the identities used. sqrt(3) is unrelated to the specific special angles of 45° and 90° involved here. The value 1 would require the numerator and denominator to be equal, which they are not after simplification.

Common Pitfalls:
A common error is to misidentify the angles in the identities, for example swapping 59° and 31° or using sin(A - B) instead of sin(A + B). Another issue is forgetting that cos 45° equals 1/sqrt(2) and instead using an incorrect value like sqrt(2)/2 without handling the division properly. Writing each step carefully helps avoid these mistakes.

Final Answer:
The value of the given expression is sqrt(2).