Final answer:
The value of tan 5° tan 25° tan 30° tan 65° tan 85° simplifies to 1/√3 using the properties of trigonometric functions and complementary angles.
Step-by-step explanation:
The value of tan 5° tan 25° tan 30° tan 65° tan 85° can be determined using properties of trigonometric functions and complementary angles. The complementary angles, such as (5° and 85°) or (25° and 65°), have tangents that are reciprocals of each other. Moreover, tan 30° is a special angle whose value is 1/√3. Combining these properties:
tan 5° × tan 85° = 1 (since tan 85° = cot 5°) and tan 25° × tan 65° = 1 (since tan 65° = cot 25°)
Therefore, the expression simplifies to:
tan 5° × tan 25° × tan 30° × tan 65° × tan 85° = 1 × 1 × tan 30° = 1/√3.
The answer is C) 1/√3.