For `x,y,z,t in R`, if `sin^(-1)x+cos^(-1)y+sec^(-1)z ge t^(2)-sqrt(2pi)*t+3pi`, then the value of `tan^(-1)x+tan^(-1)y+tan^(-1)z+tan^(-1)( sqrt((2)/(pi))t)` is __________.

Answer

To solve the problem, we need to analyze the given inequality and find the value of the expression involving inverse tangent functions.

### Step 1: Understanding the Inequality
We start with the inequality:
\[ \sin^{-1} x + \cos^{-1} y + \sec^{-1} z \geq t^2 - \sqrt{2\pi} t + 3\pi \]

### Step 2: Using Inverse Trigonometric Identities
Recall the following identities:
1. \(\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}\)
2. \(\sec^{-1} z = \cos^{-1} \left(\frac{1}{z}\right)\) for \(z \geq 1\)

### Step 3: Finding the Minimum Values
- The minimum value of \(\sin^{-1} x\) is \(-\frac{\pi}{2}\) (when \(x = -1\)).
- The minimum value of \(\cos^{-1} y\) is \(0\) (when \(y = 1\)).
- The minimum value of \(\sec^{-1} z\) is \(0\) (when \(z = 1\)).

Thus, we can say:
\[ \sin^{-1} x + \cos^{-1} y + \sec^{-1} z \geq -\frac{\pi}{2} + 0 + 0 = -\frac{\pi}{2} \]

### Step 4: Analyzing the Right-Hand Side
Now we analyze the right-hand side:
\[ t^2 - \sqrt{2\pi} t + 3\pi \]
This is a quadratic expression in \(t\). The minimum value occurs at:
\[ t = \frac{\sqrt{2\pi}}{2} \]
Substituting this back into the quadratic gives:
\[ \left(\frac{\sqrt{2\pi}}{2}\right)^2 - \sqrt{2\pi} \cdot \frac{\sqrt{2\pi}}{2} + 3\pi = \frac{2\pi}{4} - \frac{2\pi}{2} + 3\pi = \frac{\pi}{2} - \pi + 3\pi = \frac{3\pi}{2} \]

### Step 5: Setting Up the Inequality
Now we have:
\[ -\frac{\pi}{2} \geq \frac{3\pi}{2} \]
This inequality does not hold, which means the left-hand side must be greater than or equal to the right-hand side.

### Step 6: Finding the Value of the Expression
Now we need to find:
\[ \tan^{-1} x + \tan^{-1} y + \tan^{-1} z + \tan^{-1} \left(\sqrt{\frac{2}{\pi}} t\right) \]

Since \(\tan^{-1} x\), \(\tan^{-1} y\), and \(\tan^{-1} z\) can take values between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), we can analyze the maximum possible value of this expression.

### Final Step: Conclusion
From the earlier analysis, we can conclude that:
\[ \tan^{-1} x + \tan^{-1} y + \tan^{-1} z + \tan^{-1} \left(\sqrt{\frac{2}{\pi}} t\right) = \frac{\pi}{2} \]

Thus, the final answer is:
\[ \boxed{\frac{\pi}{2}} \]