Thread: Expression manipulation

$\displaystyle p = R^{j\theta}$ is a complex number where $\displaystyle R<1$. * is complex conjugation.

$\displaystyle A_1 = \frac{1}{(1-p^*p^{-1})(1-p^{-1})}$
$\displaystyle A_2 = \frac{1}{(1-pp*^{-1})(1-p*^{-1})}$
$\displaystyle A_3 = \frac{1}{(1-p)(1-p*)}$

Can anyone give a hint at how I can manipulate this expression:

$\displaystyle H_1 = \frac{A_1}{A_3}p^n + \frac{A_2}{A_3}p*^n$

into this expression

$\displaystyle H_2 = \frac{R^{n+2}\sin(\theta(n+1)) - R^{n+1}\sin(\theta(n+2))}{\sin\theta}$

n is discrete and takes on values 0,1,2,...

I have looked at quite a few trigonometric identities and I have tried to work backwards going from $\displaystyle H_2$ to $\displaystyle H_1$ but without any luck. I have verified by simulation that the two expressions are indeed identical

Arrrgh, finally figured it out.