If $\cot α = 1$ and $\sec β = - \frac{5}{3},$ where $π < α < \frac{3 π}{2}$ and $\frac{π}{2} < β < π,$ then the value of $\tan (α + β)$ and the quadrant in which $α + β$ lies, respectively are

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$- \frac{1}{7}$ and $I V^{th}$ quadrant

7 and $I^{st}$ quadrant

-7 and $I V^{th}$ quadrant

$\frac{1}{7}$ and $I^{st}$ quadrant

answer is A.

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Detailed Solution

$\cot α = 1 \Rightarrow \tan α = 1$
$\sec β = - \frac{5}{3} \Rightarrow \tan β = - \frac{4}{3}$
$∴ \tan (α + β) = \frac{\tan α + \tan β}{1 - \tan α \tan β} = \frac{1 - \frac{4}{3}}{1 + \frac{4}{3}} = - \frac{1}{7}$

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