If `(tan^(-1) x)^2 + (cot^(-1) x)^2 = (5 pi^2) /8` find x.

Explanation

To solve the equation \((\tan^{-1} x)^2 + (\cot^{-1} x)^2 = \frac{5\pi^2}{8}\), we will follow these steps:

### Step 1: Use the identity for \(\cot^{-1} x\)
We know that:
\[ \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x \]
Substituting this into the equation, we have:
\[ (\tan^{-1} x)^2 + \left(\frac{\pi}{2} - \tan^{-1} x\right)^2 = \frac{5\pi^2}{8} \]

### Step 2: Expand the equation
Expanding the squared term:
\[ (\tan^{-1} x)^2 + \left(\frac{\pi^2}{4} - \pi \tan^{-1} x + (\tan^{-1} x)^2\right) = \frac{5\pi^2}{8} \]
This simplifies to:
\[ 2(\tan^{-1} x)^2 - \pi \tan^{-1} x + \frac{\pi^2}{4} = \frac{5\pi^2}{8} \]

### Step 3: Rearranging the equation
Now, let's rearrange the equation:
\[ 2(\tan^{-1} x)^2 - \pi \tan^{-1} x + \frac{\pi^2}{4} - \frac{5\pi^2}{8} = 0 \]
To combine the constants, we convert \(\frac{\pi^2}{4}\) to have a common denominator:
\[ \frac{\pi^2}{4} = \frac{2\pi^2}{8} \]
Thus, the equation becomes:
\[ 2(\tan^{-1} x)^2 - \pi \tan^{-1} x + \left(\frac{2\pi^2}{8} - \frac{5\pi^2}{8}\right) = 0 \]
This simplifies to:
\[ 2(\tan^{-1} x)^2 - \pi \tan^{-1} x - \frac{3\pi^2}{8} = 0 \]

### Step 4: Let \(y = \tan^{-1} x\)
Let \(y = \tan^{-1} x\). The equation then becomes:
\[ 2y^2 - \pi y - \frac{3\pi^2}{8} = 0 \]

### Step 5: Solve the quadratic equation
Using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
- Here, \(a = 2\), \(b = -\pi\), and \(c = -\frac{3\pi^2}{8}\).
\[ y = \frac{\pi \pm \sqrt{(-\pi)^2 - 4 \cdot 2 \cdot \left(-\frac{3\pi^2}{8}\right)}}{2 \cdot 2} \]
Calculating the discriminant:
\[ b^2 - 4ac = \pi^2 + 6\pi^2 = 7\pi^2 \]
Thus:
\[ y = \frac{\pi \pm \sqrt{7\pi^2}}{4} = \frac{\pi \pm \sqrt{7}\pi}{4} = \frac{\pi(1 \pm \sqrt{7})}{4} \]

### Step 6: Determine valid solutions for \(y\)
Since \(y = \tan^{-1} x\) must be in the range \(-\frac{\pi}{2} < y < \frac{\pi}{2}\), we check the two possible values:
1. \(y = \frac{\pi(1 + \sqrt{7})}{4}\) (this is greater than \(\frac{\pi}{2}\) and not valid)
2. \(y = \frac{\pi(1 - \sqrt{7})}{4}\) (this is negative and valid)

### Step 7: Find \(x\)
Now, we find \(x\):
\[ x = \tan\left(\frac{\pi(1 - \sqrt{7})}{4}\right) \]

### Conclusion
Thus, the value of \(x\) is:
\[ x = \tan\left(\frac{\pi(1 - \sqrt{7})}{4}\right) \]