Find the value of the following inverse trigonometric expression:
`cos^(-1)(1/(sqrt(2))("cos"(9pi)/10-"sin"(9pi)/10))`

Explanation

To find the value of the expression \( \cos^{-1}\left(\frac{1}{\sqrt{2}} \left( \cos\left(\frac{9\pi}{10}\right) - \sin\left(\frac{9\pi}{10}\right) \right)\right) \), we can follow these steps:

### Step 1: Simplify the expression inside the inverse cosine
We start with the expression:
\[ y = \cos^{-1}\left(\frac{1}{\sqrt{2}} \left( \cos\left(\frac{9\pi}{10}\right) - \sin\left(\frac{9\pi}{10}\right) \right)\right) \]
We know that \( \frac{1}{\sqrt{2}} \) is equal to \( \cos\left(\frac{\pi}{4}\right) \) and \( \sin\left(\frac{\pi}{4}\right) \).

### Step 2: Rewrite the expression using cosine and sine
Using the cosine of a difference formula:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
We can express \( \cos\left(\frac{9\pi}{10}\right) - \sin\left(\frac{9\pi}{10}\right) \) in terms of cosine:
\[ \cos\left(\frac{9\pi}{10}\right) - \sin\left(\frac{9\pi}{10}\right) = \cos\left(\frac{\pi}{4}\right) \cos\left(\frac{9\pi}{10}\right) - \sin\left(\frac{\pi}{4}\right) \sin\left(\frac{9\pi}{10}\right) \]
This can be rewritten as:
\[ \cos\left(\frac{9\pi}{10} + \frac{\pi}{4}\right) \]

### Step 3: Find the angle
Now we need to compute:
\[ \frac{9\pi}{10} + \frac{\pi}{4} = \frac{36\pi}{40} + \frac{10\pi}{40} = \frac{46\pi}{40} = \frac{23\pi}{20} \]
Thus, we have:
\[ y = \cos^{-1}\left(\frac{1}{\sqrt{2}} \cos\left(\frac{23\pi}{20}\right)\right) \]

### Step 4: Use the properties of inverse cosine
The expression simplifies to:
\[ y = \cos^{-1}\left(\cos\left(\frac{23\pi}{20}\right)\right) \]
Since \( \frac{23\pi}{20} \) is greater than \( \pi \) but less than \( 2\pi \), we can use the property:
\[ \cos^{-1}(\cos x) = 2\pi - x \quad \text{for } x \in ( \pi, 2\pi ) \]
Thus:
\[ y = 2\pi - \frac{23\pi}{20} \]

### Step 5: Calculate the final value
Now, we find:
\[ y = \frac{40\pi}{20} - \frac{23\pi}{20} = \frac{17\pi}{20} \]

### Final Answer
The value of the expression is:
\[ \frac{17\pi}{20} \]