Question

Prove that: $\tan {20}^{\circ }\tan {35}^{\circ }\tan {45}^{\circ }\tan {55}^{\circ }\tan {70}^{\circ }=1$

Answer 1

Given: $\tan {20}^0\tan {35}^0\tan {45}^0\tan {55}^0\tan {70}^0$

this can be written as,

$=\tan ({90}^0-{70}^0)\tan ({90}^0-{55}^0)\tan {45}^0\tan {55}^0\tan {70}^0$

$=\cot {70}^0\cot {55}^0\tan {45}^0\tan {55}^0\tan {70}^0$ $\left[\tan \right(90-x)=\cot x]$

$=\frac{1}{\tan {70}^0}\frac{1}{\tan {55}^0}\tan {45}^0\tan {55}^0\tan {70}^0$ $[\cot x=\frac{1}{\tan x}]$

$=\tan {45}^{\circ }$

$=1$