Principle of Mathematical Induction

Principle of Mathematical Induction

Question

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.
                                 

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