A boy runs on a horizontal road with a speed of $4 \mathrm{m} {\mathrm{s}}^{-1}$ while it is raining. He sees that the rain is making an angle $\mathrm{\theta }$ with the vertical while running from west to east. However, when he runs from east to west, the angle is $\mathrm{\alpha }$. The rain is pouring down at an angle $45°$ with the vertical and at a speed of $8 \mathrm{m} {\mathrm{s}}^{-1}$ as shown in the figure. The ratio $\frac{\tan \theta }{\tan \alpha }$ is,

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.