Evaluate the exact value of the product tan 1° × tan 2° × tan 3° × ... × tan 89°, where the angles are in degrees and the product runs through all integer angles from 1° to 89°.

Introduction / Context:
This is a classic trigonometric product problem that highlights the complementary angle property of the tangent function. Instead of calculating each tangent individually, you should observe how tan θ and tan(90° - θ) relate to each other. This insight allows you to pair terms and simplify a very long product into something extremely simple, which is a common trick in competitive exam problems.

Given Data / Assumptions:

• The product is tan 1° × tan 2° × tan 3° × ... × tan 89°.
• Angles are in degrees.
• All angles from 1° to 89° are included exactly once.

Concept / Approach:
The key identity is tan(90° - θ) = cot θ, and the fact that tan θ * cot θ = 1 for all angles θ where both are defined. In the given product, terms can be paired: tan 1° with tan 89°, tan 2° with tan 88°, and so on. Each such pair multiplies to 1 because tan(90° - θ) is the reciprocal of tan θ. The only term that does not have a distinct complement in the range 1° to 89° is tan 45°, because 45° is its own complement. Since tan 45° = 1, it does not change the product.

Step-by-Step Solution:
Write the product as P = tan 1° × tan 2° × ... × tan 44° × tan 45° × tan 46° × ... × tan 89°. Notice that 89° is 90° - 1°, 88° is 90° - 2°, and so on up to 46° which is 90° - 44°. Pair terms: (tan 1° × tan 89°), (tan 2° × tan 88°), ..., (tan 44° × tan 46°). Use tan(90° - θ) = cot θ, so tan(90° - θ) = 1 / tan θ. Therefore, tan θ × tan(90° - θ) = tan θ × (1 / tan θ) = 1 for each such pair. There are 44 such pairs from 1° through 44° combined with 89° down to 46°. The remaining middle term is tan 45°. Since tan 45° = 1, it does not change the product. Thus the total product P is 1 × 1 × ... × 1 × tan 45° = 1 × 1 = 1.

Verification / Alternative check:
As a sanity check, you might approximate a few paired values: tan 10° ≈ 0.176 and tan 80° ≈ 5.671. Their product is close to 1. Similarly, tan 30° ≈ 0.577 and tan 60° ≈ 1.732, whose product is almost exactly 1.33? Actually, tan 30° * tan 60° = (1 / sqrt(3)) * sqrt(3) = 1 exactly. These examples illustrate the general pairing pattern and give confidence that every complementary pair yields a product of 1.

Why Other Options Are Wrong:
0 would require at least one term to be zero, but tan θ is zero only when θ is a multiple of 180°, which does not occur between 1° and 89°. Infinity is not valid because tangent remains finite on (0°, 90°). The values 2 and -1 are not supported by the pairing argument, which shows that the product equals 1 exactly.

Common Pitfalls:
Some students attempt to compute many of the tangents numerically, which is impossible under exam conditions and unnecessary. Others forget the complementary angle property and treat the product as something that must grow very large. Always look for patterns in symmetrical ranges like 1° to 89°, especially involving tan θ and tan(90° - θ).

Final Answer:
The product tan 1° × tan 2° × ... × tan 89° is exactly 1.