Ex9 1

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 9.1}$

Verify that

(1)
\begin{align} g_i = \sin ((2i+1)\theta)\\ b_i = \cos ((2i+1)\theta), \end{align}

with $\sin\theta = \sqrt{t} = \sqrt{{\abs{G}}/{N}}$, is a solution to the recurrence relations of section 9.1.4.