A particle is projected up an inclined plane of inclination `beta` at na elevation `prop` to the horizontal. Show that
(a) `tan prop = cot beta + 2 tan beta`, if the particle strikes the plane at right angles
(b) `tan prop = 2 tan beta`, if the particle strikes the plane horizontally.

Explanation

To solve the problem, we will analyze the motion of a particle projected up an inclined plane at an angle \( \alpha \) to the horizontal, and we will derive the required equations for two different scenarios: when the particle strikes the plane at right angles and when it strikes the plane horizontally.

### Part (a): Show that \( \tan \alpha = \cot \beta + 2 \tan \beta \) if the particle strikes the plane at right angles.

1. **Identify the Components of Motion**:
- The inclined plane makes an angle \( \beta \) with the horizontal.
- The particle is projected at an angle \( \alpha \) to the horizontal.
- The velocity components can be broken down into two directions: along the incline and perpendicular to the incline.
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