Question

If $\tan A=\frac{1}{3}$ and $\sec B=\frac{13}{5}$ where $\pi <A<\frac{3\pi }{2}$, $\frac{3\pi }{2}<B<2\pi$, then $\cot (A+B)$ is equal to

• A. $-\frac{27}{31}$
• B. $\frac{27}{31}$
• C. $-\frac{31}{27}$
• D. $\frac{31}{27}$

Answer 1

The correct option is A $-\frac{27}{31}$

$\tan A=\frac{1}{3}\Rightarrow \cot A=3$
$\sec B=\frac{13}{5}\Rightarrow \cos B=\frac{5}{13}\Rightarrow \sin B=-\frac{12}{13}\Rightarrow \cot B=-\frac{5}{12}$
$(\because \frac{3\pi }{2}<B<2\pi )$
$\cot (A+B)=\frac{\cot A\cot B-1}{\cot B+\cot A}$
$=\frac{3\times (-\frac{5}{12})-1}{-\frac{5}{12}+3}=-\frac{27}{31}$