Induction Help ! (1 Viewer)

YashC3 · Apr 30, 2010

Ive come across this question quite a bit but I am still unsure of how to go about proving it.

Prove by induction that cos (x + n pie) = (-1)^n cosx for all positive integers of n.

Pwnage101 · Apr 30, 2010

I'll only do the inductive step:
$\textup{Assume the result is true for}\; n=k\; \textup{i.e.}\; \cos \left ( x+k\pi \right )=\left ( -1 \right )^{k}\cos x\\ \textup{Required to prove true for}\; n=k+1\; \textup{i.e.}\;\cos \left ( x+(k+1)\pi \right )=\left ( -1 \right )^{k+1}\cos x\\ \begin{align*} \textup{Now}\cos \left ( x+(k+1)\pi \right )&=\cos \left ( x+k\pi +\pi \right )\\ &=\cos \left ( \left \{ x+k\pi \right \}+\pi \right )\\ &=\cos\left ( x+k\pi \right ) \cos \pi -\sin\left ( x+k\pi \right ) \sin \pi \\ &=\cos\left ( x+k\pi \right ) \left ( -1 \right )-\sin\left ( x+k\pi \right ) \left ( 0 \right )\\ &=-\cos\left ( x+k\pi \right ) \\ &=-\left ( -1 \right )^{k}\cos x\; \textup{from our assumption}\\ &=\left ( -1 \right )^{k+1}\cos x\\ \; \end{align*}$

Drongoski · Apr 30, 2010

To prove: cos(x + n.pi) = (-1)ⁿcos x

$For\ n=1:\ \ LHS\ =\ cos(x+1\pi )\ =\ -cos\ x\ =\ (-1)^{1}cos\ x\ =\ \ RHS\\ \\ \therefore \ \ formula\ holds \ for\ n=1\\ \\ Assume\ formula\ holds\ for\ n=k>=1\\ \\ i.e.\ \ cos(x+k\pi )\ =\ (-1^{k})cos\ x\\ \\\therefore \ \ cos(x+[k+1]\pi )\ =\ cos(x+k\pi + \pi )\ =\ -cos(x+k\pi )\ =\\ \\ \ -[(-1)^{k}cos\ x]\ =\ (-1)^{k+1}cos\ x\\ \\ i.e.\ \ if\ true\ for\ n=k\ \ it \ is\ true\ for\ n=k+1\\ \\ \therefore \ \ By\ mathematical \ induction\ it\ holds\ for\ all\ integer\ n\ >=1$

YashC3 · Apr 30, 2010

Cheers

Induction Help ! (1 Viewer)

YashC3

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Drongoski

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YashC3

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