Question

The value of cot 5× cot 15 × cot25× cot35 × cot45 × cot55×cot65×cot75×cot85

Find the value of cot .?

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{cot\,5^\circ{\times}cot\,15^\circ{\times}cot\,25^\circ{\times}cot\,35^\circ{\times}cot\,45^\circ{\times}cot\,55^\circ{\times}cot\,65^\circ{\times}cot\,75^\circ{\times}cot\,85^\circ}$

$\underline{\textbf{To find:}}$

$\textsf{The value of}$

$\mathsf{cot\,5^\circ{\times}cot\,15^\circ{\times}cot\,25^\circ{\times}cot\,35^\circ{\times}cot\,45^\circ{\times}cot\,55^\circ{\times}cot\,65^\circ{\times}cot\,75^\circ{\times}cot\,85^\circ}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{Concept used:}}$

$\mathsf{1.\,cot\theta=tan(90^\circ-\theta)}$

$\mathsf{2.\,cot\theta=\dfrac{1}{tan\theta}}$

$\mathsf{Consider,}$

$\mathsf{cot\,5^\circ{\times}cot\,15^\circ{\times}cot\,25^\circ{\times}cot\,35^\circ{\times}cot\,45^\circ{\times}cot\,55^\circ{\times}cot\,65^\circ{\times}cot\,75^\circ{\times}cot\,85^\circ}$

$\textsf{This can be written as}$

$\mathsf{=cot\,5^\circ{\times}cot\,15^\circ{\times}cot\,25^\circ{\times}cot\,35^\circ{\times}1{\times}tan\,35^\circ{\times}tan\,25^\circ{\times}tan\,15^\circ{\times}tan\,5^\circ}$

$\mathsf{=\dfrac{1}{tan\,5^\circ}{\times}\dfrac{1}{tan\,15^\circ}{\times}\dfrac{1}{tan\,25^\circ}{\times}\dfrac{1}{tan\,35^\circ}{\times}1{\times}tan\,35^\circ{\times}tan\,25^\circ{\times}tan\,15^\circ{\times}tan\,5^\circ}$

$\mathsf{=1}$

$\implies\boxed{\mathsf{cot\,5^\circ{\times}cot\,15^\circ{\times}cot\,25^\circ{\times}cot\,35^\circ{\times}cot\,45^\circ{\times}cot\,55^\circ{\times}cot\,65^\circ{\times}cot\,75^\circ{\times}cot\,85^\circ=1}}$