Find the value of cot (tan-1 α + cot-1 α)

To find the value of \(\cot (\tan^{-1} \alpha + \cot^{-1} \alpha)\), we start by defining some variables for clarity.

Let:
\[ \theta = \tan^{-1} \alpha \]
and
\[ \phi = \cot^{-1} \alpha \]

Thus:
\[ \alpha = \tan \theta \]
and
\[ \alpha = \cot \phi \]

Since \(\cot \phi = \frac{1}{\tan \phi}\), we have:
\[ \alpha = \frac{1}{\tan \phi} \implies \tan \phi = \frac{1}{\alpha} \]

Next, we need to find \(\theta + \phi\). To do this, we use the fact that \(\tan \theta = \alpha\) and \(\tan \phi = \frac{1}{\alpha}\). Using the angle addition formula for tangent, we get:
\[ \tan(\theta + \phi) = \frac{\tan \theta + \tan \phi}{1 - \tan \theta \tan \phi} \]

Substituting \(\tan \theta = \alpha\) and \(\tan \phi = \frac{1}{\alpha}\):
\[ \tan(\theta + \phi) = \frac{\alpha + \frac{1}{\alpha}}{1 - \alpha \cdot \frac{1}{\alpha}} = \frac{\alpha + \frac{1}{\alpha}}{1 - 1} \]

Since the denominator is zero, \(\tan(\theta + \phi)\) is undefined, which implies that \(\theta + \phi = \frac{\pi}{2}\) because \(\tan \frac{\pi}{2}\) is undefined.

Now, we want to find \(\cot (\theta + \phi)\):
\[ \cot (\theta + \phi) = \cot \left( \frac{\pi}{2} \right) \]

Since \(\cot \frac{\pi}{2} = 0\):
\[ \cot (\tan^{-1} \alpha + \cot^{-1} \alpha) = 0 \]

Thus, the value is:
\[ \boxed{0} \]

Would you like details on any part of this solution, or do you have any other questions? Here are some related questions you might find interesting:

1. Find the value of \(\sin (\tan^{-1} x + \cot^{-1} x)\).
2. Simplify \(\tan (\sin^{-1} x + \cos^{-1} x)\).
3. Solve for \(\alpha\) in \(\cot (\sin^{-1} \alpha + \cos^{-1} \alpha) = 1\).
4. Evaluate \(\cot (\sec^{-1} x + \csc^{-1} x)\).
5. Find the value of \(\tan (\cos^{-1} x + \sin^{-1} x)\).

**Tip:** When working with inverse trigonometric functions, remember their domains and ranges to avoid undefined expressions or incorrect simplifications.

Discover how to find the value of cot (tan^-1 α + cot^-1 α) with a detailed step-by-step solution. This problem involves trigonometric identities and properties of inverse trigonometric functions, providing insights for advanced high school students.

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