10 grade maths> Given x + x -1 = 2 cos y, x 2 + x -2 = 2 ...

Given x + x-1 = 2 cos y,x2 + x-2 = 2 cos (2y), Prove that xn + x-n = 2 cos (ny) by mathematical Induction.

Michael , 10 Years ago

Grade 11

1 Answers

Sumit Majumdar

Last Activity: 10 Years ago

Dear student,

Let us assume

$x^{k}+x^{-k}=2cos\left ( ky \right )$

and thus

$x^{k-1}+x^{-k+1}=2cos\left ( \left (k-1 \right )y \right )$

So, we need to prove it for
$x^{k+1}+x^{-k-1}=2cos\left ( \left (k+1 \right )y \right )$

Proof $\left (x^{k}+x^{-k} \right )\times \left (x^{1}+x^{-1} \right )= 4cos\left ( 2ky)cos\left ( 2y \right ) \Rightarrow \left (x^{k+1}+x^{-k-1} \right )= 2\left ( cos\left ( 2k+1 \right )y \right +cos\left ( 2k-1 \right )y \right)-\left (x^{-k+1}+x^{k-1} \right )= 2\left ( cos\left ( 2k+1 \right )y \right \right)$ :

$\left (x^{k}+x^{-k} \right )\times \left (x^{1}+x^{-1} \right )=4cos\left ( 2ky)cos\left ( 2y \right ) \Rightarrow \left (x^{k+1}+x^{-k-1} \right )=2\left ( cos\left ( 2k+1 \right )y \right +cos\left ( 2k-1 \right )y \right)-\left (x^{-k+1}+x^{k-1} \right )=2\left ( cos\left ( 2k+1 \right )y \right \right)$