Solve for x cot inverse x plus tan inverse 8 is equal to pi by 2.

To solve the equation 

\[
\cot^{-1}(x) + \tan^{-1}(8) = \frac{\pi}{2},
\]

we will use a trigonometric identity.

### Step-by-Step Solution

1. **Recall the trigonometric identity:**

   \[
   \cot^{-1}(x) + \tan^{-1}(x) = \frac{\pi}{2}.
   \]

   This identity holds true for all real \(x > 0\).

2. **Apply the identity to our equation:**

   Given that \(\cot^{-1}(x) + \tan^{-1}(8) = \frac{\pi}{2}\), we can compare this to the identity:

   \[
   \cot^{-1}(x) = \tan^{-1}(8).
   \]

3. **Equating the inverse functions:**

   The equality \(\cot^{-1}(x) = \tan^{-1}(8)\) implies that \(x = \cot(\tan^{-1}(8))\).

4. **Simplify \(\cot(\tan^{-1}(8))\):**

   To find \(\cot(\tan^{-1}(8))\), we need to express it in terms of a right triangle. If \(\theta = \tan^{-1}(8)\), then:

   \[
   \tan(\theta) = 8 = \frac{\text{opposite}}{\text{adjacent}}.
   \]

   We can consider a right triangle where the opposite side is 8 and the adjacent side is 1. The hypotenuse, \(h\), can be calculated using the Pythagorean theorem:

   \[
   h = \sqrt{8^2 + 1^2} = \sqrt{64 + 1} = \sqrt{65}.
   \]

   The cotangent is the reciprocal of the tangent:

   \[
   \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{1}{8}.
   \]

   Hence,

   \[
   \cot(\tan^{-1}(8)) = \frac{1}{8}.
   \]

5. **Conclusion:**

   So, \(x = \frac{1}{8}\).

### Final Answer

\[
\boxed{x = \frac{1}{8}}.
\]

Would you like more details or have any questions? 

Here are 5 related questions for further exploration:

1. What are the properties of inverse trigonometric functions?
2. How can we solve equations involving both trigonometric and inverse trigonometric functions?
3. What is the geometric interpretation of \(\tan^{-1}(x)\) and \(\cot^{-1}(x)\)?
4. Can we derive other identities involving inverse trigonometric functions?
5. How can we use the Pythagorean theorem to simplify expressions involving trigonometric ratios?

**Tip:** Remember that understanding the relationships between trigonometric functions and their inverses can greatly simplify complex problems.

Learn how to solve the equation cot^{-1}(x) + tan^{-1}(8) = \frac{\pi}{2} step-by-step with detailed explanations. This problem involves trigonometric identities and the Pythagorean theorem. Suitable for high school students.

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