Question

Evaluate \( \cot 12^\circ \cdot \cot 38^\circ \cdot \cot 52^\circ \cdot \cot 60^\circ \cdot \cot 78^\circ \):

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

Hint:

Use identities and complementary angles to simplify products of trigonometric functions.

Answer 1

The Correct Option is A

We are given the product \( \cot 12^\circ \cdot \cot 38^\circ \cdot \cot 52^\circ \cdot \cot 60^\circ \cdot \cot 78^\circ \). Using the identity \( \cot \theta = \tan (90^\circ - \theta) \), we can simplify the angles: \[ \cot 12^\circ = \tan 78^\circ, \quad \cot 38^\circ = \tan 52^\circ. \] Thus, the product becomes: \[ \tan 78^\circ \cdot \tan 52^\circ \cdot \cot 60^\circ = \tan 78^\circ \cdot \tan 52^\circ \cdot \frac{1}{\tan 30^\circ}. \] Since \( \tan 30^\circ = \frac{1}{\sqrt{3}} \), the product simplifies to: \[ \tan 78^\circ \cdot \tan 52^\circ \cdot \sqrt{3} = 1. \] Therefore, the answer is \( \boxed{1} \).