`tan(cot^(-1)x)` is equal to

Options

A. `(pi)/(2)-x`
B. `cot(tan^(-1)x)`
C. tanx
D. None of these

Answer

To solve the expression \( \tan(\cot^{-1} x) \), we can follow these steps:

### Step 1: Understand the relationship between inverse trigonometric functions
We know that:
\[ \tan(\cot^{-1} x) = \tan\left(\frac{\pi}{2} - \tan^{-1} x\right) \]
This is because \( \cot^{-1} x \) can be expressed as \( \frac{\pi}{2} - \tan^{-1} x \).

### Step 2: Apply the tangent identity
Using the identity \( \tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta) \), we can rewrite the expression:
\[ \tan(\cot^{-1} x) = \cot(\tan^{-1} x) \]

### Step 3: Find the value of \( \cot(\tan^{-1} x) \)
The cotangent of an angle is the reciprocal of the tangent. Therefore:
\[ \cot(\tan^{-1} x) = \frac{1}{\tan(\tan^{-1} x)} \]
Since \( \tan(\tan^{-1} x) = x \), we have:
\[ \cot(\tan^{-1} x) = \frac{1}{x} \]

### Final Answer
Thus, we conclude that:
\[ \tan(\cot^{-1} x) = \frac{1}{x} \]

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