 Solution

Choose $0<\theta <\pi /2$ with $\sin \theta =\sqrt{p}$. Then $1-2p=\cos 2\theta$ and $2\sqrt{p(1-p)}=\sin 2\theta$. The matrix found in part 2 is a rotation by $2\theta$ in the ordered good-bad plane, with the bad coordinate listed first. The initial coordinate vector is $(\cos \theta ,\sin \theta {)}^T$.

After $r$ rotations its angle from the bad axis is $(2r+1)\theta$. Equivalently, multiplication of the displayed matrix and the addition formulas for sine and cosine prove the assertion by induction from $r=0$. HenceThe Born rule therefore gives good-output probability ${\sin }^2((2r+1)\theta )$. This is the geometric mechanism of amplitude amplification: each coherent iteration rotates the already existing good amplitude toward the good axis.