If `tanx+tany=25` and `cotx+coty=30` then the value of `tan(x+y)` is

Options

A. `100`
B. `150`
C. `250`
D. `200`

Explanation

To solve the problem, we have the equations:

1. \( \tan x + \tan y = 25 \)
2. \( \cot x + \cot y = 30 \)

We need to find the value of \( \tan(x+y) \).

### Step 1: Rewrite cotangent in terms of tangent
We know that:
\[ \cot x = \frac{1}{\tan x} \quad \text{and} \quad \cot y = \frac{1}{\tan y} \]
So we can rewrite the second equation as:
\[ \frac{1}{\tan x} + \frac{1}{\tan y} = 30 \]

### Step 2: Combine the cotangent equation
Multiplying through by \( \tan x \tan y \) gives us:
\[ \tan y + \tan x = 30 \tan x \tan y \]
Substituting \( \tan x + \tan y = 25 \) into this equation, we have:
\[ 25 = 30 \tan x \tan y \]

### Step 3: Solve for \( \tan x \tan y \)
Rearranging gives:
\[ \tan x \tan y = \frac{25}{30} = \frac{5}{6} \]

### Step 4: Use the tangent addition formula
We can use the formula for \( \tan(x+y) \):
\[ \tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \]
Substituting the known values:
\[ \tan(x+y) = \frac{25}{1 - \frac{5}{6}} \]

### Step 5: Simplify the denominator
Calculating the denominator:
\[ 1 - \frac{5}{6} = \frac{1}{6} \]

### Step 6: Substitute back into the tangent addition formula
Now substituting back, we have:
\[ \tan(x+y) = \frac{25}{\frac{1}{6}} = 25 \times 6 = 150 \]

### Final Answer
Thus, the value of \( \tan(x+y) \) is:
\[ \boxed{150} \]