Question

If $\tan A+\tan B=p$ and $\cot A+\cot B=q$ then $\cot (A+B)$ is

• A. $\frac{p-q}{pq}$
• B. $\frac{q-p}{pq}$
• C. $\frac{pq}{p+q}$
• D. $\frac{pq}{p-q}$

Answer 1

Answer: (B)

$\frac{q-p}{pq}$

$\tan A+\tan B=p$ and $\cot A+\cot B=q$

$\Rightarrow \frac{1}{\tan A}+\frac{1}{\tan B}=q$

$\Rightarrow \frac{\tan A+\tan B}{\tan A\tan B}=q$

$\Rightarrow \frac{p}{\tan A\tan B}=q$

$\Rightarrow \tan A\tan B=\frac{p}{q}$

$\cot (A+B)=\frac{1}{\tan (A+B)}$

$=\frac{1}{\frac{\tan A+\tan B}{1-\tan A\tan B}}$

$=\frac{1-\tan A\tan B}{\tan A+\tan B}$

$=\frac{1-\frac{p}{q}}{p}$

$=\frac{q-p}{pq}$