Step 1: Identify the Angles: First angle $\theta_1 = \pi/8 = 22.5^\circ$.
Second angle $\theta_2 = \pi/2 - \pi/8 = 3\pi/8 = 67.5^\circ$.
Step 2: Relate Heights to Angles: The formula for maximum height is $H = \frac{u^2 \sin^2 \theta}{2g}$.
Since $u$ and $g$ are constant, $H \propto \sin^2 \theta$.
$$\frac{H_1}{H_2} = \frac{\sin^2 \theta_1}{\sin^2 \theta_2} = \frac{\sin^2(\pi/8)}{\sin^2(3\pi/8)}$$
Note that $\sin(3\pi/8) = \cos(\pi/8)$, so:
$$\frac{H_1}{H_2} = \frac{\sin^2(\pi/8)}{\cos^2(\pi/8)} = \tan^2(\pi/8)$$
Step 3: Calculate $\tan^2(\pi/8)$: Using the identity $\tan^2(\theta/2) = \frac{1 - \cos \theta}{1 + \cos \theta}$ for $\theta = \pi/4$:
$$\tan^2(\pi/8) = \frac{1 - \cos(\pi/4)}{1 + \cos(\pi/4)} = \frac{1 - 1/\sqrt{2}}{1 + 1/\sqrt{2}} = \frac{\sqrt{2} - 1}{\sqrt{2} + 1}$$
Multiplying by conjugate: $(\sqrt{2}-1)^2 = 3 - 2\sqrt{2} \approx 0.1715$.