Question

\( \tan 5^\circ \cdot \tan 25^\circ \cdot \tan 30^\circ \cdot \tan 65^\circ \cdot \tan 85^\circ \) is:

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

Hint:

The identity \( \tan x \cdot \tan (90^\circ - x) = 1 \) is useful for such problems.

Answer 1

The Correct Option is A

Using the property:
\[ \tan x \cdot \tan (90^\circ - x) = 1 \] Pairing the terms:
\[ \tan 5^\circ \cdot \tan 85^\circ = 1 \] \[ \tan 25^\circ \cdot \tan 65^\circ = 1 \] And \( \tan 30^\circ = \frac{1}{\sqrt{3}} \).
Thus:
\[ 1 \times 1 \times \frac{1}{\sqrt{3}} \times \sqrt{3} = 1 \]