Problem 9 - Entrance Test

The expression is a product of tangents of angles from 1° to 89°. We use the complementary angle identity: tan (90° - θ) = cot θ. Consider pairs of terms from the beginning and end: tan 1° * tan 89° = tan 1° * tan (90° - 1°) = tan 1° * cot 1° = 1. tan 2° * tan 88° = tan 2° * tan (90° - 2°) = tan 2° * cot 2° = 1. This pattern continues up to tan 44° * tan 46° = 1. The middle term is tan 45°, which is 1. So, the product is (tan 1° tan 89°)(tan 2° tan 88°)...(tan 44° tan 46°) * tan 45°. = (1)(1)...(1) * 1 = 1.