Question

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

Answer 1

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

Hence, The correct answer is option (a)