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We know that the Initial state $|\psi\rangle$ can be represented as $\sin\frac{\theta}{2}|\chi\rangle + \cos\frac{\theta}{2}|\xi\rangle$. We can prove the result $G^R|\psi\rangle = \sin\frac{(2R+1)\theta}{2}|\chi\rangle + \cos\frac{(2R+1)\theta}{2}|\xi\rangle$ by Induction Base Case When $R=0$, $G^R=G^0=I$ and $2R+1=2\times0+1=1$. Thus \$G^0|\psi\rangle = \...