If: `tan^(-1)(1/3) + tan^(-1)( 3/4) - tan^(-1)(x/3) =0`, then: x=

Options

A. `7/3`
B. 3
C. `11/3`
D. `13/3`

Answer

To solve the equation \( \tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{3}{4}\right) - \tan^{-1}\left(\frac{x}{3}\right) = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate the inverse tangent term:
\[ \tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{3}{4}\right) = \tan^{-1}\left(\frac{x}{3}\right) \]

**Hint:** Remember that if \( A + B = C \), then you can express it as \( A + B - C = 0 \).

### Step 2: Using the Formula for the Sum of Inverse Tangents
We can use the formula for the sum of two inverse tangents:
\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \]
where \( ab < 1 \).

In our case, let \( a = \frac{1}{3} \) and \( b = \frac{3}{4} \):
\[ \tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{3}{4}\right) = \tan^{-1}\left(\frac{\frac{1}{3} + \frac{3}{4}}{1 - \frac{1}{3} \cdot \frac{3}{4}}\right) \]

**Hint:** Ensure that the product \( ab \) is less than 1 before applying the formula.

### Step 3: Calculating the Numerator and Denominator
Now, we calculate the numerator and denominator:
1. **Numerator**:
\[ \frac{1}{3} + \frac{3}{4} = \frac{4 + 9}{12} = \frac{13}{12} \]

2. **Denominator**:
\[ 1 - \left(\frac{1}{3} \cdot \frac{3}{4}\right) = 1 - \frac{1}{4} = \frac{3}{4} \]

So we have:
\[ \tan^{-1}\left(\frac{13/12}{3/4}\right) = \tan^{-1}\left(\frac{13 \cdot 4}{12 \cdot 3}\right) = \tan^{-1}\left(\frac{52}{36}\right) = \tan^{-1}\left(\frac{13}{9}\right) \]

**Hint:** Simplifying fractions can help in finding a clearer result.

### Step 4: Setting the Two Inverse Tangents Equal
Now we have:
\[ \tan^{-1}\left(\frac{13}{9}\right) = \tan^{-1}\left(\frac{x}{3}\right) \]

**Hint:** If \( \tan^{-1}(A) = \tan^{-1}(B) \), then \( A = B \).

### Step 5: Equating the Arguments
From the equality of the inverse tangents, we can equate the arguments:
\[ \frac{13}{9} = \frac{x}{3} \]

### Step 6: Solving for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{13}{9} \cdot 3 = \frac{39}{9} = \frac{13}{3} \]

Thus, the final answer is:
\[ \boxed{\frac{13}{3}} \]