Question

Find the value of $tan(\alpha +\beta )$, given that
$cot\alpha =\frac{1}{2},\alpha \in \left(\pi ,\frac{3\pi }{2}\right)$ and $sec\beta =\frac{-5}{3},\beta \in \left(\frac{\pi }{2},\pi \right)$.

• A. $1/11$
• B. $2/11$
• C. $5/11$
• D. $3/11$

Answer 1

Answer: (B)

Given, $cot\alpha =\frac{1}{2}$,
$\Rightarrow tan\alpha =2$ and $sec\beta =\frac{-5}{3}$
Then, $tan\beta =\sqrt{se{c}^2\beta -1}$
$\Rightarrow tan\beta =\pm \frac{4}{3}$
But, $tan\beta =\frac{-4}{3}$ $(\because tan\beta$ is -$ve$ in $II$ quadrant$)$
$\therefore tan\left(\alpha +\beta \right)=\frac{tan\alpha +tan\beta }{1-tan\alpha \cdot tan\beta }$
$=\frac{2+\left(-\frac{4}{3}\right)}{1-\left(2\right)\left(\frac{-4}{3}\right)}=\frac{2}{11}$