Question

Geometric mean of $\tan 1^{\circ}, \tan 2^{\circ}, \ldots \ldots . \ldots, \tan 89^{\circ}$ is

• A. $\frac{1}{89}$
• B. $1$
• C. $\frac{1}{3}$
• D. $\sqrt{3}$

Answer 1

Answer: (B)

$\tan 1^{\circ} \cdot \tan 2^{\circ} \ldots \tan 45^{\circ} \cdot \tan 46^{\circ} \ldots \tan 88^{\circ} \cdot \tan 89^{\circ}$
$\tan 1^{\circ} \cdot \tan 2^{\circ} \ldots \tan 44^{\circ} \cdot \tan 45^{\circ} \tan \left(90-44^{\circ}\right) \ldots \tan \left(90^{\circ}-1^{\circ}\right)$
$\tan 1^{\circ} \cdot \tan 2^{\circ} \ldots \tan 44^{\circ} \tan 45^{\circ} \operatorname{Cot} 44^{\circ} \cdot \operatorname{Cot} 43^{\circ} \ldots \operatorname{Cot} 1^{\circ}$
$\tan 1^{\circ} \cdot \tan 2^{\circ} \ldots \tan 44^{\circ} 1 \frac{1}{\tan 44^{\circ}} \frac{1}{\tan 43^{\circ}} \cdots s \frac{1}{\tan 1^{\circ}}=1$