Results 1 to 2 of 2

Math Help - prove the following

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    9

    prove the following

    prove
    tan(x-(a/2)) = [(k-1)tan(a/2)]/(k+1)
    given that
    sinx=k sin(a-x)
    my ans is slightly different
    on rhs denominator i get 1+ktan^2(a/2)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1
    Hello IIT 2010
    Quote Originally Posted by IIT 2010 View Post
    prove
    tan(x-(a/2)) = [(k-1)tan(a/2)]/(k+1)
    given that
    sinx=k sin(a-x)
    my ans is slightly different
    on rhs denominator i get 1+ktan^2(a/2)
    \sin x = k\sin(a-x)

    \Rightarrow \sin x = k(\sin a\cos x - \cos a\sin x)

    \Rightarrow \sin x(1+k\cos a) = k\sin a\cos x

    \Rightarrow \tan x = \frac{k\sin a}{1+k\cos a}

    = \frac{\dfrac{2kt}{1+t^2}}{1+\dfrac{k(1-t^2)}{1+t^2}}, where t =\tan \tfrac12a

     =\frac{2kt}{1+t^2+k(1-t^2)}

    \Rightarrow \tan(x-\tfrac12a)=\frac{\tan x -\tan\tfrac12a}{1+\tan x \tan\tfrac12a}

    =\frac{\dfrac{2kt}{1+t^2+k(1-t^2)} -t}{1+\dfrac{2kt^2}{1+t^2+k(1-t^2)}}

    =\frac{2kt-t\Big(1+t^2+k(1-t^2)\Big)}{1+t^2+k(1-t^2)+2kt^2}

    =\frac{t(2k-1-t^2-k+kt^2)}{1+t^2+k-kt^2+2kt^2}

    =\frac{t\Big((k-1)+t^2(k-1)\Big)}{(k+1)+t^2(k+1)}

    =\frac{t(k-1)(1+t^2)}{(k+1)(1+t^2)}

    =\frac{(k-1)\tan\tfrac12a}{(k+1)}

    Grandad
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove a/b and a/c then a/ (3b-7c)
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 23rd 2010, 05:20 PM
  2. prove,,,
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 1st 2010, 09:02 AM
  3. Prove |w + z| <= |w| +|z|
    Posted in the Algebra Forum
    Replies: 3
    Last Post: February 28th 2010, 05:44 AM
  4. Replies: 2
    Last Post: August 28th 2009, 02:59 AM
  5. How to prove that n^2 + n + 2 is even??
    Posted in the Algebra Forum
    Replies: 3
    Last Post: November 30th 2008, 01:24 PM

Search Tags


/mathhelpforum @mathhelpforum