If $space space cot space straight alpha space equals space 1 half$ and $sec space straight beta space equals space minus 5 over 3$ where $straight pi less than straight alpha less than fraction numerator 3 straight pi over denominator 2 end fraction$ and $straight pi over 2 less than straight beta less than straight pi comma$ find the

value of $tan left parenthesis straight alpha plus straight beta right parenthesis.$ State the quadrant in which $space space left parenthesis straight alpha plus straight beta right parenthesis$ terminates.

Answer

   $cotα space equals space 1 half$
$rightwards double arrow$ $space space tanα space equals space 2$
$secβ space equals space minus 5 over 3$
We know that $space space sec squared straight beta space equals space 1 plus tan squared straight beta$
$rightwards double arrow$ $open parentheses fraction numerator negative 5 over denominator 3 end fraction close parentheses squared space equals space 1 plus tan squared straight beta$
$rightwards double arrow$    $tan squared straight beta space equals space 25 over 9 minus 1 space equals space 16 over 9$
$rightwards double arrow$ $tanβ space equals space minus 4 over 3$ [∵ $tanβ$ is -ve as $straight beta$ lies in IInd quadrant]
$rightwards double arrow$    $space space space space tan left parenthesis straight alpha plus straight beta right parenthesis space equals space fraction numerator tanα plus tanβ over denominator 1 minus tanα space tanβ end fraction space equals space fraction numerator 2 minus begin display style 4 over 3 end style over denominator 1 minus 2 open parentheses begin display style fraction numerator negative 4 over denominator 3 end fraction end style close parentheses end fraction space equals space fraction numerator begin display style fraction numerator 6 minus 4 over denominator 3 end fraction end style over denominator begin display style fraction numerator 3 plus 8 over denominator 3 end fraction end style end fraction equals space 2 over 11$
Now,    $straight pi less than straight alpha less than fraction numerator 3 straight pi over denominator 2 end fraction$ and $straight pi over 2 less than straight beta less than straight pi$
$rightwards double arrow$ $fraction numerator 3 straight pi over denominator 2 end fraction less than straight alpha plus straight beta less than fraction numerator 5 straight pi over denominator 2 end fraction$
We know that $tan left parenthesis straight alpha plus straight beta right parenthesis space is space plus ve$ in Ist and IIIrd quadrant.
∴ $straight alpha plus straight beta$ lies in 1st quadrant.