Question

find the value of tan(alpha+beta), given that cot alpha=1/2, alpha€(pi,3pie/2), sec beta=-5/3, beta€(pie/2,pie). Also state the quadrant in which alpha+beta terminates
$cot \alpha = \frac{1}{2} \: ( lies \: in \: 3rd \: quadrant)| \\ sec \beta = - \frac{5}{3} (lies \: in \: 2nd \: quadrant) \\ tan \alpha = \frac{1}{cot \alpha } = 2 \\ {tan}^{2} \beta = {sec}^{2} \beta - 1 \\ {tan}^{2} \beta = \frac{25}{9} - 1 \\ {tan}^{2} \beta = \frac{16}{9} \\ tan \beta = \sqrt{ \frac{16}{9} } \\ tan \beta = \frac{ - 4}{3} \: (tan \beta \: is \: negative \: in \: 2nd \: quadrant) \\ tan( \alpha + \beta ) = \frac{tan \alpha + tan \beta }{1 - tan \alpha tan \beta }$
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