Question

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

• A. 0
• B. 1
• C. Infinity
• D. none

Answer 1

Answer: (B)

1

Given,

$\tan 1^0\tan 2^0\tan 3^0......\tan {89}^0$

$=\tan ({90}^0-1^0)\tan ({90}^0-2^0)\tan ({90}^0-3^0)......\tan {89}^0$

$=\cot {89}^0\cot {88}^0\cot {87}^0......\tan {89}^0$ [$\because \tan ({90}^0-\theta )=\cot \theta )$]

$=\cot {89}^0\tan {89}^0\cot {88}^0\tan {88}^0......\tan {45}^0$ [$because\tan \theta \times \cot \theta =1\tan {45}^{\circ }=1$]

$=1$