Question

If $\cot \theta =\frac{1}{2}$ and $\sec \varphi =-\frac{5}{3}$, where $\theta \in (\pi ,\frac{3\pi }{2})$ and $\varphi \in (\frac{\pi }{2},\pi )$, then the value of $\cot (\theta -\varphi )$ is

• A. $2$
• B. $-\frac{1}{2}$
• C. $\frac{1}{2}$
• D. $\frac{3}{2}$

Answer 1

The correct option is B $-\frac{1}{2}$

$\cot \theta =\frac{1}{2}\sec \varphi =-\frac{5}{3}\Rightarrow \tan \varphi =\sqrt{{\sec }^2\varphi -1}\tan \varphi =\pm \frac{4}{3}$
As $\varphi \in (\frac{\pi }{2},\pi )\Rightarrow \tan \varphi =-\frac{4}{3}\Rightarrow \cot \varphi =-\frac{3}{4}$
$\cot (\theta -\varphi )=\frac{\cot \theta \cot \varphi +1}{\cot \varphi -\cot \theta }=\frac{\frac{1}{2}\times (-\frac{3}{4})+1}{-\frac{3}{4}-\frac{1}{2}}=-\frac{1}{2}$