Vectors a and b are given and there is an angle θ between them. We are also given that vectors a, b, and a-b subtend angles α and β with vector a. We need to find tan α * tan β.

Step 1: Understanding the problem

Let's first understand the given information and the problem statement:

- Vectors a and b: These are two vectors given in the problem statement.
- Angle θ: This is the angle between vectors a and b.
- Vectors a, b, and a-b subtend angles α and β with vector a: This means that when we draw the vectors a, b, and a-b from the same point, the angles α and β are formed between vector a and vectors b and (a-b) respectively.

Step 2: Visualizing the scenario

To better understand the problem, let's visualize the scenario:

- Draw vector a starting from the origin.
- Draw vector b starting from the end point of vector a.
- Draw vector (a-b) starting from the end point of vector b.

Now, we have a triangle formed by vectors a, b, and (a-b). Let's denote the length of vector a as |a|, the length of vector b as |b|, and the length of vector (a-b) as |a-b|. We also have the angles α and β.

Step 3: Applying trigonometry

We can apply trigonometry to solve this problem. Let's use the law of cosines to find the length of vector (a-b):

|a-b|^2 = |a|^2 + |b|^2 - 2|a||b|cos(θ)

Now, we have the lengths of all three vectors: |a|, |b|, and |a-b|. We can use these lengths to calculate the tangents of angles α and β.

tan(α) = |b|sin(α) / (|a| - |b|cos(α))

tan(β) = |b|sin(β) / (|a| + |b|cos(β))

Step 4: Finding tan α * tan β

To find tan α * tan β, we can multiply the two equations:

tan α * tan β = (|b|sin(α) / (|a| - |b|cos(α))) * (|b|sin(β) / (|a| + |b|cos(β)))

Simplifying this expression will give us the final answer.

Step 5: Simplifying the expression

tan α * tan β = (|b|^2sin(α)sin(β)) / ((|a| - |b|cos(α)) * (|a| + |b|cos(β)))

Step 6: Conclusion

By following the steps above, we have derived the expression for tan α * tan β using the given information. Now, you can substitute the values of |a|, |b|, α, and β to calculate the final answer.

B square sin square theta divided by a square minus b square cos square theta