Question

Evaluate the following product: \[ \tan 1^\circ \times \tan 2^\circ \times \tan 3^\circ \times \cdots \times \tan 89^\circ \]

• A. \( \sqrt{3} \)
• B. 1
• C. \( \sqrt{2} \)
• D. 2

Hint:

For products of tangent functions involving complementary angles, use the identity \( \tan x \times \tan(90^\circ - x) = 1 \) to simplify the expression.

Answer 1

The Correct Option is B

Step 1: Recognize the identity.
The product of tangent values from \( \tan 1^\circ \) to \( \tan 89^\circ \) has a well-known trigonometric identity. Specifically, the identity: \[ \tan x \times \tan(90^\circ - x) = 1. \] This identity pairs up terms such that each pair of angles (e.g., \( \tan 1^\circ \) and \( \tan 89^\circ \)) multiplies to 1.
Step 2: Simplifying the product.
By applying this identity to all pairs, we see that the entire product simplifies to 1. Thus, the correct answer is 1.

Step 3: Conclusion.
Therefore, the correct answer is option (B) 1.