Question

State whether the following statement is true or false.

${\cot }^{-1}7+{\cot }^{-1}8+{\cot }^{-1}18={\cot }^{-1}3$

• A. True
• B. False

Answer 1

The correct option is A True

LHS :

${\cot }^{-1}7+{\cot }^{-1}8+{\cot }^{-1}18$

$={\tan }^{-1}\frac{1}{7}+{\tan }^{-1}\frac{1}{8}+{\tan }^{-1}\frac{1}{18}$

$={\tan }^{-1}\left(\frac{\frac{1}{7}+\frac{1}{8}}{1-\frac{1}{7}\times \frac{1}{8}}\right)+{\tan }^{-1}\frac{1}{18}$

$={\tan }^{-1}\frac{15}{55}+{\tan }^{-1}\frac{1}{18}$

$={\tan }^{-1}\frac{3}{11}+{\tan }^{-1}\frac{1}{18}$

$={\tan }^{-1}\left(\frac{\frac{3}{11}+\frac{1}{18}}{1-\frac{3}{11}\times \frac{1}{18}}\right)$

$={\tan }^{-1}\left(\frac{65}{195}\right)={\tan }^{-1}\frac{1}{3}={\cot }^{-1}3$