Question

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

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

Answer 1

Answer: (B)

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