Question

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

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

Answer 1

The correct option is B $\frac{q-p}{pq}$

$\tan A+\tan B=p$ ………..$\left(1\right)$

$\cot A+\cot B=q$ ………….$\left(2\right)$

From $\left(1\right)$

$\frac{1}{\cot A}+\frac{\cot B}{=}p$

$\frac{\cot A+\cot B}{\cot A\cot B}=p$

$\frac{q}{\cot A\cot B}=p$

$\cot A\cot B=\frac{q}{p}$

$\cot (A+B)=\frac{\cot A\cot B-1}{\cot A+\cot B}=\frac{\frac{q}{p}-1}{q}$

$\cot (A+B)=\frac{q-p}{pq}$.