A man observes when he has climbed up `1/3` of the length of an inclined ladder, placed against a wall, the angular depression of an object on the floor is `alphadot` When he climbs the ladder completely, the angleof depression is `beta` . If the inclination of the ladder to the floor is `theta,` then prove that `cottheta=(3cotbeta-cotalpha)/2`

Explanation

To solve the problem step-by-step, we will analyze the situation described and use trigonometric relationships to derive the required formula.

### Step 1: Understand the Setup
We have a ladder inclined at an angle \( \theta \) with the ground. The man climbs \( \frac{1}{3} \) of the length of the ladder and observes an object on the floor at an angle of depression \( \alpha \). When he climbs the ladder completely, the angle of depression is \( \beta \).

### Step 2: Define Variables
Let:
- \( L \) = Length of the ladder
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