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