The value of `tan(tan^(-1)5+"cot"^(-1)(1)/(3))` is :

Options

A. `(4)/(7)`
B. `-(4)/(7)`
C. `(3)/(7)`
D. Not defined

Answer

To solve the problem \( \tan\left(\tan^{-1}(5) + \cot^{-1}\left(\frac{1}{3}\right)\right) \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \tan\left(\tan^{-1}(5) + \cot^{-1}\left(\frac{1}{3}\right)\right) \]

### Step 2: Use the cotangent identity
Recall that \( \cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right) \). Therefore, we can rewrite \( \cot^{-1}\left(\frac{1}{3}\right) \) as:
\[ \cot^{-1}\left(\frac{1}{3}\right) = \tan^{-1}(3) \]

### Step 3: Substitute back into the expression
Now we can substitute this back into our expression:
\[ \tan\left(\tan^{-1}(5) + \tan^{-1}(3)\right) \]

### Step 4: Use the addition formula for tangent
We can use the formula for the tangent of a sum:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
In our case, \( A = \tan^{-1}(5) \) and \( B = \tan^{-1}(3) \), so:
\[ \tan\left(\tan^{-1}(5) + \tan^{-1}(3)\right) = \frac{\tan(\tan^{-1}(5)) + \tan(\tan^{-1}(3))}{1 - \tan(\tan^{-1}(5)) \tan(\tan^{-1}(3))} \]

### Step 5: Simplify using the values of tangent
Since \( \tan(\tan^{-1}(5)) = 5 \) and \( \tan(\tan^{-1}(3)) = 3 \), we substitute these values:
\[ = \frac{5 + 3}{1 - (5)(3)} = \frac{8}{1 - 15} = \frac{8}{-14} \]

### Step 6: Simplify the fraction
Now, simplifying \( \frac{8}{-14} \):
\[ = \frac{4}{-7} = -\frac{4}{7} \]

### Final Answer
Thus, the value of \( \tan\left(\tan^{-1}(5) + \cot^{-1}\left(\frac{1}{3}\right)\right) \) is:
\[ -\frac{4}{7} \]