Results 1 to 8 of 8

Thread: Proving by induction

  1. #1
    Junior Member
    Joined
    Mar 2011
    Posts
    33

    Proving by induction

    If sinx cannot = 0, use mathematical induction to show that

    $\displaystyle cosx \cdot cos2x \cdot cos4x... cos2^{n-1} x = \frac{sin2^n x}{2^n sinx}$ for every integer $\displaystyle n \geq 1$

    So for the induction step what I have so far is:

    Assume n=k is true, show true for n=k+1

    $\displaystyle cosx \cdot cos2x \cdot cos4x... cos2^{k-1} x \cdot cos2^{(k+1)-1}x = \frac{sin2^{k+1} x}{2^{k+1} sinx}$

    $\displaystyle \frac{sin2^{k}x}{2^{k}sinx} \cdot cos2^{k}x = \frac{sin2^{k+1}x}{2^{k+1}sinx}$

    $\displaystyle \frac{(sin2^{k}x)(cos2^{k}x)}{2^{k}sinx} = \frac{sin2^{k+1}x}{2^{k+1}sinx}$

    Assuming any of this is correct so far, I am having trouble showing that both sides are equal from here on.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,577
    Thanks
    790

    Re: Proving by induction

    Note that sin(2x) = 2sin(x)cos(x).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2011
    Posts
    33

    Re: Proving by induction

    So then I'll have:

    $\displaystyle \frac{(sin2^{k}x)(cos2^{k}x)}{2^{k}sinx} = \frac{2sin^{k+1}x cos^{k+1}x}{2^{k+1}sinx}$

    Is that what you meant I should do next? Where do I go from here then?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,577
    Thanks
    790

    Re: Proving by induction

    Quote Originally Posted by BobRoss View Post
    So then I'll have:

    $\displaystyle \frac{(sin2^{k}x)(cos2^{k}x)}{2^{k}sinx} = \frac{2sin^{k+1}x cos^{k+1}x}{2^{k+1}sinx}$

    Is that what you meant I should do next?
    No, the right-hand side is wrong.. We have $\displaystyle \sin(2^{k+1}x)=\sin(2\cdot2^kx)=2\sin(2^kx)\cos(2^ kx)$.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Mar 2011
    Posts
    33

    Re: Proving by induction

    Quote Originally Posted by emakarov View Post
    No, the right-hand side is wrong.. We have $\displaystyle \sin(2^{k+1}x)=\sin(2\cdot2^kx)=2\sin(2^kx)\cos(2^ kx)$.
    Why does $\displaystyle \sin(2^{k+1}x)=\sin(2\cdot2^kx)$

    ?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member DIOGYK's Avatar
    Joined
    Aug 2012
    From
    Norway
    Posts
    39

    Re: Proving by induction

    Quote Originally Posted by BobRoss View Post
    Why does $\displaystyle \sin(2^{k+1}x)=\sin(2\cdot2^kx)$

    ?
    For example:
    $\displaystyle a^2*a^3=a^{2+3}=a^{5}.$.
    Similarly,
    $\displaystyle \sin(2^1*2^kx)=\sin(2^{k+1}x)$
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Mar 2011
    Posts
    33

    Re: Proving by induction

    Oh okay that makes sense. Then I can do the same thing with the term $\displaystyle 2^{k+1}sinx$ to get $\displaystyle 2 \cdot 2^{k}sinx$ And doing that makes the left side equal the right side, and that proves the equation, correct?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,577
    Thanks
    790

    Re: Proving by induction

    Quote Originally Posted by BobRoss View Post
    Oh okay that makes sense. Then I can do the same thing with the term $\displaystyle 2^{k+1}sinx$ to get $\displaystyle 2 \cdot 2^{k}sinx$ And doing that makes the left side equal the right side, and that proves the equation, correct?
    Yes.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proving inequality through induction
    Posted in the Advanced Math Topics Forum
    Replies: 18
    Last Post: May 22nd 2012, 09:13 AM
  2. Proving Complete Induction
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: Jun 10th 2011, 01:13 PM
  3. proving (by induction)
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Aug 30th 2009, 04:50 PM
  4. few problems on proving using induction
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: Mar 22nd 2009, 02:00 PM
  5. Need help proving!! Mathematical Induction
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Oct 15th 2008, 06:38 PM

Search Tags


/mathhelpforum @mathhelpforum