Simplify [sin (90 – 10θ) – cos (π – 6θ)] / [cos (π/2 – 10θ) – sin (π – 6θ)] and express the result in terms of θ.

Introduction / Context:
This problem uses complementary and supplementary angle identities in trigonometry. Angles like 90 degrees and π are common in such identities, and the expression is constructed so that standard formulas simplify it to a basic function of θ. The goal is to recognize these patterns and reduce the complex fraction.

Given Data / Assumptions:

• Expression E = [sin(90 - 10θ) - cos(π - 6θ)] / [cos(π/2 - 10θ) - sin(π - 6θ)].
• θ is a real angle for which the denominators and trigonometric functions are defined.
• We want E in a simple form in terms of θ.

Concept / Approach:
We use the standard identities: sin(90 - α) = cos α, cos(π - α) = -cos α, cos(π/2 - α) = sin α, sin(π - α) = sin α. By applying these, the expression becomes a ratio of sums and differences of sine and cosine terms in 10θ and 6θ. Then we use sum to product identities for sine and cosine to simplify the ratio to a single cotangent function.

Step-by-Step Solution:
First, simplify sin(90 - 10θ). Using sin(90 - α) = cos α, we get sin(90 - 10θ) = cos 10θ. Next, simplify cos(π - 6θ). Using cos(π - α) = -cos α, we get cos(π - 6θ) = -cos 6θ. So the numerator becomes cos 10θ - ( -cos 6θ ) = cos 10θ + cos 6θ. Now simplify cos(π/2 - 10θ). Using cos(π/2 - α) = sin α, we get cos(π/2 - 10θ) = sin 10θ. Simplify sin(π - 6θ). Using sin(π - α) = sin α, we get sin(π - 6θ) = sin 6θ. Thus the denominator is sin 10θ - sin 6θ. Now use the sum to product identity for cosine: cos C + cos D = 2 cos((C + D)/2) cos((C - D)/2). So cos 10θ + cos 6θ = 2 cos 8θ cos 2θ. For the denominator, use sin C - sin D = 2 cos((C + D)/2) sin((C - D)/2). So sin 10θ - sin 6θ = 2 cos 8θ sin 2θ. Therefore E = [2 cos 8θ cos 2θ] / [2 cos 8θ sin 2θ] = cos 2θ / sin 2θ = cot 2θ.

Verification / Alternative check:
Choose a suitable value such as θ = 10 degrees and calculate the original expression numerically using trigonometric tables or a calculator. Then compute cot 2θ, which is cot 20 degrees. Both results match, confirming that the simplification is correct.

Why Other Options Are Wrong:
tan 2θ: This is the reciprocal of cot 2θ and would come from inverting the numerator and denominator by mistake.
cot θ and cot 3θ: These suggest incorrect combinations of angles or misusing identities for double or triple angle functions.
tan θ: This ignores the presence of both 10θ and 6θ and is not consistent with the sum to product reduction we used.

Common Pitfalls:
Errors often occur in remembering the signs in cos(π - α) = -cos α and in applying the complementary angle identities. Another typical issue is mixing up the sum to product identities for sine and cosine. Careful substitution and stepwise simplification ensure that the final answer is correct.

Final Answer:
The simplified value of the expression is cot 2θ.