Value of `"tan" (1)/(3) (tan^(-1)x + "tan"^(-1) (1)/(x))`

Options

A. `(1)/(sqrt2)`
B. `(1)/(sqrt3)`
C. `sqrt3`
D. `-sqrt3`

Answer

To solve the given problem, we need to find the value of

\[ \tan\left(\frac{1}{3} \left(\tan^{-1} x + \tan^{-1} \frac{1}{x}\right)\right). \]

### Step 1: Use the identity for inverse tangent
We know that

\[ \tan^{-1} x + \tan^{-1} \frac{1}{x} = \frac{\pi}{2} \]

for \( x > 0 \). This is because \( \tan^{-1} x \) and \( \tan^{-1} \frac{1}{x} \) are complementary angles.

### Step 2: Substitute the identity into the expression
Substituting this identity into our expression, we have:

\[ \tan\left(\frac{1}{3} \left(\frac{\pi}{2}\right)\right). \]

### Step 3: Simplify the expression
This simplifies to:

\[ \tan\left(\frac{\pi}{6}\right). \]

### Step 4: Calculate the value of \(\tan\left(\frac{\pi}{6}\right)\)
We know that

\[ \tan\left(\frac{\pi}{6}\right) = \tan(30^\circ) = \frac{1}{\sqrt{3}}. \]

### Final Answer
Thus, the value of

\[ \tan\left(\frac{1}{3} \left(\tan^{-1} x + \tan^{-1} \frac{1}{x}\right)\right) = \frac{1}{\sqrt{3}}. \]

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