if `(3sin^(-1)x+pix-pi)^(2)+{sin(cos^(-1)((x)/(5)))}^(2)-2sin(cos^(-1)((x)/(5)))=-1` then which of the following is/are Correct?

Explanation

To solve the equation

\[ (3 \sin^{-1}(x) + \pi x - \pi)^2 + \left(\sin\left(\cos^{-1}\left(\frac{x}{5}\right)\right)\right)^2 - 2 \sin\left(\cos^{-1}\left(\frac{x}{5}\right)\right) = -1, \]

we will break it down step by step.

### Step 1: Simplify the equation
We can rewrite the equation as follows:

\[ (3 \sin^{-1}(x) + \pi x - \pi)^2 + \left(\sin\left(\cos^{-1}\left(\frac{x}{5}\right)\right)\right)^2 - 2 \sin\left(\cos^{-1}\left(\frac{x}{5}\right)\right) + 1 = 0. \]

### Step 2: Analyze the first part
Let’s denote:

\[ A = 3 \sin^{-1}(x) + \pi x - \pi. \]

Thus, we have:

\[ A^2 + \left(\sin\left(\cos^{-1}\left(\frac{x}{5}\right)\right)\right)^2 - 2 \sin\left(\cos^{-1}\left(\frac{x}{5}\right)\right) + 1 = 0. \]

### Step 3: Solve for the first equation
For \(A^2 = 0\), we need:

\[ 3 \sin^{-1}(x) + \pi x - \pi = 0. \]

Rearranging gives:

\[ 3 \sin^{-1}(x) = \pi - \pi x. \]

### Step 4: Solve for \(x\)
Dividing both sides by 3:

\[ \sin^{-1}(x) = \frac{\pi - \pi x}{3}. \]

Taking the sine of both sides:

\[ x = \sin\left(\frac{\pi - \pi x}{3}\right). \]

### Step 5: Analyze the second part
Now, we analyze the second part:

\[ \sin\left(\cos^{-1}\left(\frac{x}{5}\right)\right) = \sqrt{1 - \left(\frac{x}{5}\right)^2}. \]

Thus, we can rewrite the second part of the equation:

\[ \left(\sqrt{1 - \left(\frac{x}{5}\right)^2}\right)^2 - 2\sqrt{1 - \left(\frac{x}{5}\right)^2} + 1 = 0. \]

This simplifies to:

\[ 1 - \left(\frac{x}{5}\right)^2 - 2\sqrt{1 - \left(\frac{x}{5}\right)^2} + 1 = 0. \]

### Step 6: Solve the quadratic equation
This can be rearranged to:

\[ 2 - \left(\frac{x}{5}\right)^2 - 2\sqrt{1 - \left(\frac{x}{5}\right)^2} = 0. \]

Let \(y = \sqrt{1 - \left(\frac{x}{5}\right)^2}\), then we have:

\[ 2 - \left(\frac{x}{5}\right)^2 - 2y = 0. \]

### Step 7: Find solutions
From the first equation, we found \(x = \frac{1}{2}\).

From the second part, we find that:

\[ \frac{x}{5} = 0 \implies x = 0. \]

### Conclusion
Since the two equations yield different values for \(x\) (0 and \(\frac{1}{2}\)), there are no common solutions. Therefore, the correct option is:

**D: No solution.**