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Math Help - proving identities....

  1. #1
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    proving identities....

    Hey.

    a) By using the subsitution t= tan 1/2 x
    prove that cosecx- cotx= tan 1/2x

    b) use this result to show that tan15= 2- sqrt 3

    THANK YOU!
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  2. #2
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    Hello Oasis1993
    Quote Originally Posted by Oasis1993 View Post
    Hey.

    a) By using the subsitution t= tan 1/2 x
    prove that cosecx- cotx= tan 1/2x

    b) use this result to show that tan15= 2- sqrt 3

    THANK YOU!
    If t =\tan\tfrac12x, then
    \sin x = \frac{2t}{1-t^2} and \tan x = \frac{2t}{1+t^2}.
    So:
    \csc x -\cot x=\frac{1+t^2}{2t}-\frac{1-t^2}{2t}
    =\frac{1+t^2-(1-t^2)}{2t}

    =\frac{2t^2}{2t}

    =t

    =\tan\tfrac12x
    Put x = 15^o, noting that \tan30^o=\frac{1}{\sqrt3} and \sin30^o=\frac12, and you're done.

    Grandad
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