If $\cot \alpha=\frac{1}{2}$ and $\sec \beta=-\frac{5}{3}$ where a $\in\left(\pi, \frac{3 \pi}{2} ight)$ and $\beta \in\left(\frac{\pi}{2}, \pi ight)$, then $ an (\alpha+\beta)$ has the value

Question

If $\cot \alpha=\frac{1}{2}$ and $\sec \beta=-\frac{5}{3}$ where a $\in\left(\pi, \frac{3 \pi}{2}ight)$ and $\beta \in\left(\frac{\pi}{2}, \piight)$, then $	an (\alpha+\beta)$ has the value

Aicharya · Accepted Answer

$\cot \alpha=\frac{1}{2} \Rightarrow 	an \alpha=2$ $\sec \beta=\frac{-5}{3} \Rightarrow 	an \beta=-\frac{4}{3}$ Now, $	an (\alpha+\beta)=\frac{	an 	heta+	an \beta}{1-	an \alpha \cdot 	an \beta}=\frac{2+\left(\frac{-4}{3}ight)}{1-2 	imes\left(\frac{-4}{3}ight)}=\frac{\frac{6-4}{3}}{\frac{3+8}{3}}=\frac{2}{11}$

If $\cot \alpha=\frac{1}{2}$ and $\sec \beta=-\frac{5}{3}$ where a $\in\left(\pi, \frac{3 \pi}{2}\right)$…

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