If `tan(x+y)=33`, and `x= tan^(-1)3`, then: y=

Options

A. 0.3
B. `tan^(-1)(1.3)`
C. `tan^(-1)(0.3)`
D. ` tan^(-1)(1/18)`

Answer

To solve the problem step by step, we start with the given information and apply the properties of inverse trigonometric functions.

### Step-by-Step Solution:

1. **Given Information**:
- We have \( \tan(x + y) = 33 \)
- We also know \( x = \tan^{-1}(3) \)

2. **Substituting the Value of \( x \)**:
- Substitute \( x \) into the equation:
\[ \tan(\tan^{-1}(3) + y) = 33 \]

3. **Using the Addition Formula for Tangent**:
- The formula for \( \tan(a + b) \) is:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
- Here, let \( a = \tan^{-1}(3) \) and \( b = y \). Therefore, we have:
\[ \tan(\tan^{-1}(3) + y) = \frac{\tan(\tan^{-1}(3)) + \tan(y)}{1 - \tan(\tan^{-1}(3)) \tan(y)} \]
- Since \( \tan(\tan^{-1}(3)) = 3 \), we can rewrite the equation as:
\[ \frac{3 + \tan(y)}{1 - 3\tan(y)} = 33 \]

4. **Cross-Multiplying**:
- Cross-multiply to eliminate the fraction:
\[ 3 + \tan(y) = 33(1 - 3\tan(y)) \]
- Expanding the right side:
\[ 3 + \tan(y) = 33 - 99\tan(y) \]

5. **Rearranging the Equation**:
- Move all terms involving \( \tan(y) \) to one side:
\[ \tan(y) + 99\tan(y) = 33 - 3 \]
- This simplifies to:
\[ 100\tan(y) = 30 \]

6. **Solving for \( \tan(y) \)**:
- Divide both sides by 100:
\[ \tan(y) = \frac{30}{100} = 0.3 \]

7. **Finding \( y \)**:
- Now, we can find \( y \):
\[ y = \tan^{-1}(0.3) \]

### Final Answer:
Thus, the value of \( y \) is:
\[ y = \tan^{-1}(0.3) \]