Question

The value of $\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }\dots \tan {89}^{\circ }$ is

• A. $1$
• B. $\frac{1}{2}$
• C. $0$
• D. not defined

Answer 1

The correct option is A $1$

Evaluating the given expression by using relevant transformation formula

Given expression :

$\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }\dots tan{89}^{\circ }$

$=\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }\dots \tan {45}^{\circ }\dots \tan {87}^{\circ }\tan {88}^{\circ }\tan {89}^{\circ }$

$=\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }\dots 1\dots \tan ({90}^{\circ }-3^{\circ })\tan ({90}^{\circ }-2^{\circ })\tan ({90}^{\circ }-1^{\circ })$

$=\tan 1^{\circ }\tan 2^{\circ }\tan 3^{\circ }\dots 1\dots \dots \cot 3^{\circ }\cot 2^{\circ }\cot 1^{\circ }$

$[\because \tan ({90}^{\circ }-\theta )=\cot \theta ]$

$=1$

$[\because \tan \theta \cot \theta =1]$

Hence, the option (B) is correct.