Question

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

• A. $1$
• B. $0$
• C. $\infty$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $1$

$\tan 1^{\circ }\tan 2^{\circ }\cdots \tan {88}^{\circ }\tan {89}^{\circ }$
$=\tan 1^{\circ }\tan 2^{\circ }\cdots \tan {45}^{\circ }\cdots \tan (90-2{)}^{\circ }\tan (90-1{)}^{\circ }$
$=\tan 1^{\circ }\tan 2^{\circ }\cdots \tan {45}^{\circ }\cdots \cot 2^{\circ }\cot 1^{\circ }$
$=\left(\tan 1^{\circ }\cot 1^{\circ }\right)\left(\tan 2^{\circ }\cot 2^{\circ }\right)\cdots \tan {45}^{\circ }$
$=1\times 1\times \cdots \times 1=1$