Evaluate : `tan[2tan^(-1)(1/2)-cot^(-1)3]`

Answer

To evaluate the expression \( \tan\left[2\tan^{-1}\left(\frac{1}{2}\right) - \cot^{-1}(3)\right] \), we can follow these steps:

### Step 1: Use the double angle formula for tangent
We know that:
\[ 2\tan^{-1}(x) = \tan^{-1}\left(\frac{2x}{1-x^2}\right) \]
Let \( x = \frac{1}{2} \). Then:
\[ 2\tan^{-1}\left(\frac{1}{2}\right) = \tan^{-1}\left(\frac{2 \cdot \frac{1}{2}}{1 - \left(\frac{1}{2}\right)^2}\right) = \tan^{-1}\left(\frac{1}{1 - \frac{1}{4}}\right) = \tan^{-1}\left(\frac{1}{\frac{3}{4}}\right) = \tan^{-1}\left(\frac{4}{3}\right) \]

### Step 2: Rewrite the expression
Now substitute back into the original expression:
\[ \tan\left[2\tan^{-1}\left(\frac{1}{2}\right) - \cot^{-1}(3)\right] = \tan\left[\tan^{-1}\left(\frac{4}{3}\right) - \cot^{-1}(3)\right] \]

### Step 3: Use the identity for tangent of a difference
Using the identity:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Let \( A = \tan^{-1}\left(\frac{4}{3}\right) \) and \( B = \cot^{-1}(3) \). We know that:
\[ \tan B = \cot^{-1}(3) \implies \tan B = \frac{1}{3} \]
Now we can apply the identity:
\[ \tan\left[\tan^{-1}\left(\frac{4}{3}\right) - \cot^{-1}(3)\right] = \frac{\frac{4}{3} - \frac{1}{3}}{1 + \frac{4}{3} \cdot \frac{1}{3}} = \frac{\frac{4 - 1}{3}}{1 + \frac{4}{9}} = \frac{\frac{3}{3}}{\frac{13}{9}} = \frac{1}{\frac{13}{9}} = \frac{9}{13} \]

### Step 4: Final result
Thus, the final result is:
\[ \tan\left[2\tan^{-1}\left(\frac{1}{2}\right) - \cot^{-1}(3)\right] = \frac{9}{13} \]