Question

Prove that $\tan {10}^o\tan {15}^o\tan {75}^o\tan {80}^o=1$

Answer 1

Consider $\tan {10}^0$ and $\tan {15}^0$ which can be rewritten as follows:

$\tan {10}^0=\tan {({90}^0-80}^0)=\cot {80}^0=\frac{1}{\tan {80}^0}$

$\tan {15}^0=\tan {({90}^0-75}^0)=\cot {75}^0=\frac{1}{\tan {75}^0}$

$(\because \cot x=\frac{1}{\tan x},\tan ({90}^0-x)=\cot x)$

Therefore, $\tan {10}^0\tan {15}^0\tan {75}^0\tan {80}^0$ can be evaluated as shown below:

$\tan {10}^0\tan {15}^0\tan {75}^0\tan {80}^0=\frac{1}{\tan {80}^0}\times \frac{1}{\tan {75}^0}\times \tan {75}^0\times \tan {80}^0=\frac{\tan {75}^0}{\tan {75}^0}\times \frac{\tan {80}^0}{\tan {80}^0}=1$

Hence, $\tan {10}^0\tan {15}^0\tan {75}^0\tan {80}^0=1$.