Question

Find \[ \tan1^\circ \tan2^\circ \tan3^\circ \cdots \tan89^\circ \]

• A. \(2\sqrt6\)
• B. \(0\)
• C. \(1\)
• D. \(2\)

Hint:

In trigonometric products, always check complementary angle pairing.

Answer 1

The Correct Option is C

Use the identity: \[ \tan\theta \tan(90^\circ-\theta)=1 \] Pair terms: \[ \tan1^\circ \tan89^\circ=1 \] \[ \tan2^\circ \tan88^\circ=1 \] \[ \tan3^\circ \tan87^\circ=1 \] Similarly all pairs become \(1\). Middle term: \[ \tan45^\circ=1 \] So: \[ \tan1^\circ \tan2^\circ \cdots \tan89^\circ=1\times1\times1\cdots\times1=1 \] Thus, \[ \boxed{1} \]