Chapter 6.3, Problem 92AYU

If $\cot \theta \text{=}-\text{2}$, find the value of:

$\cot \theta \text{+}\text{cot}\left(\theta \text{-}\pi \right)\text{+}\text{cot}\left(\theta \text{-}\text{2}\pi \right)$

To determine

To find: $\text{cot}\theta +\text{cot}\left(\theta -\pi \right)+\text{cot}\left(\theta -2\pi \right)$

Answer to Problem 92AYU

Solution:

$\text{cot}\theta +\text{cot}\left(\theta -\pi \right)+\text{cot}\left(\theta -2\pi \right)=-6$

Explanation of Solution

Given:

$\text{cot}\theta =-2$

Calculation:

$\text{cot}\theta +\text{cot}\left(\theta -\pi \right)+\text{cot}\left(\theta -2\pi \right)=\text{cot}\theta -\text{cot}\left(\pi -\theta \right)-\text{cot}\left(2\pi -\theta \right)$

since $\text{cot}\left(\theta -\pi \right)=-\text{cot}\left(\pi -\theta \right)$

$=\text{cot}\theta +\text{cot}\theta +\text{cot}\theta$, since $\text{cot}\left(\pi -\theta \right)=\text{cot}\theta and\text{cot}\left(2\pi -\theta \right)=-\text{cot}\theta$

$=-2-2-2=-6$