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

Explanation

To solve 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 will follow these steps:

### Step 1: Rewrite the expression
We start with 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 express \( \frac{1}{\sqrt{2}} \) as \( \cos\left(\frac{\pi}{4}\right) \) and \( \frac{1}{\sqrt{2}} \) as \( \sin\left(\frac{\pi}{4}\right) \). Thus, we rewrite the expression as:
\[ \cos^{-1}\left(\cos\left(\frac{\pi}{4}\right) \left(\cos\left(\frac{9\pi}{10}\right) - \sin\left(\frac{9\pi}{10}\right)\right)\right) \]

### Step 2: Use the cosine subtraction formula
Using the identity \( \cos A \cos B - \sin A \sin B = \cos(A + B) \), we can rewrite:
\[ \cos\left(\frac{\pi}{4}\right) \cos\left(\frac{9\pi}{10}\right) - \sin\left(\frac{\pi}{4}\right) \sin\left(\frac{9\pi}{10}\right) = \cos\left(\frac{\pi}{4} + \frac{9\pi}{10}\right) \]

### Step 3: Calculate the angle
Now, we calculate the angle:
\[ \frac{\pi}{4} + \frac{9\pi}{10} = \frac{5\pi}{20} + \frac{18\pi}{20} = \frac{23\pi}{20} \]
Thus, we have:
\[ \cos^{-1}\left(\cos\left(\frac{23\pi}{20}\right)\right) \]

### Step 4: Find the value of the inverse cosine
The value of \( \cos^{-1}(\cos x) \) is \( x \) if \( x \) is in the range \( [0, \pi] \). However, \( \frac{23\pi}{20} \) is greater than \( \pi \). To bring it within the range, we can use the property:
\[ \cos(2\pi - x) = \cos x \]
Thus, we can write:
\[ \frac{23\pi}{20} = 2\pi - \frac{17\pi}{20} \]
So, we have:
\[ \cos^{-1}\left(\cos\left(\frac{23\pi}{20}\right)\right) = 2\pi - \frac{23\pi}{20} = \frac{17\pi}{20} \]

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

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