Two forces P+ Q, P-Q make an angle `2alpha` with each other and their resultant makes an angle `theta` with bisector of the angle between them, then `P tan theta= Q tan alpha`.

Answer

To prove that \( P \tan \theta = Q \tan \alpha \), we can follow these steps:

### Step 1: Understand the Problem
We have two forces, \( P + Q \) and \( P - Q \), making an angle \( 2\alpha \) with each other. The resultant of these forces makes an angle \( \theta \) with the bisector of the angle between them.

### Step 2: Identify the Angles
The angle between the two forces is \( 2\alpha \). The bisector of this angle divides it into two equal angles of \( \alpha \) each. Therefore, the angle between the resultant and one of the forces can be expressed as:
- The angle between \( P + Q \) and the bisector is \( \alpha - \theta \).
- The angle between \( P - Q \) and the bisector is \( \alpha + \theta \).

### Step 3: Use the Tangent Function
Using the tangent of the angles, we can express the tangent of the angles formed:
\[ \tan(\alpha - \theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{(P - Q) \sin(2\alpha)}{(P + Q) + (P - Q) \cos(2\alpha)} \]

### Step 4: Apply the Tangent Addition Formula
We can express \( \tan(\alpha - \theta) \) using the tangent subtraction formula:
\[ \tan(\alpha - \theta) = \frac{\tan \alpha - \tan \theta}{1 + \tan \alpha \tan \theta} \]

### Step 5: Cross-Multiply and Rearrange
Cross-multiplying gives us:
\[ (P - Q) \sin(2\alpha) = (P + Q)(\tan \alpha - \tan \theta) + (P - Q) \cos(2\alpha) \tan \alpha \tan \theta \]

### Step 6: Simplify the Equation
After simplifying, we can rearrange the equation to isolate terms involving \( P \) and \( Q \):
\[ P \tan \theta = Q \tan \alpha \]

### Conclusion
This leads us to the desired result:
\[ P \tan \theta = Q \tan \alpha \]