if `cos^(-1)((1)/(sqrt(2))(cos((7pi)/(5))-sin((2pi)/(5)))=(ppi)/(q)` (where p,q are in lowest form), then value of `(q-p)` is

Explanation

To solve the problem, we need to evaluate the expression given in the question step by step.

### Step 1: Understanding the Expression
We start with the equation:
\[ \cos^{-1}\left(\frac{1}{\sqrt{2}}\left(\cos\left(\frac{7\pi}{5}\right) - \sin\left(\frac{2\pi}{5}\right)\right)\right) = \frac{p\pi}{q} \]
where \( p \) and \( q \) are integers in their lowest form.

### Step 2: Simplifying the Argument of Cosine
We know that:
\[ \sin\left(\pi + \theta\right) = -\sin\left(\theta\right) \]
So, we can express \(\sin\left(\frac{7\pi}{5}\right)\) in terms of \(\sin\left(\frac{2\pi}{5}\right)\):
\[ \sin\left(\frac{7\pi}{5}\right) = -\sin\left(\frac{2\pi}{5}\right) \]
This allows us to rewrite the expression:
\[ \cos^{-1}\left(\frac{1}{\sqrt{2}}\left(\cos\left(\frac{7\pi}{5}\right) + \sin\left(\frac{7\pi}{5}\right)\right)\right) \]

### Step 3: Using Trigonometric Identities
We can express \(\frac{1}{\sqrt{2}}\) as \(\cos\left(\frac{\pi}{4}\right)\) and \(\sin\left(\frac{\pi}{4}\right)\):
\[ \frac{1}{\sqrt{2}} = \cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) \]
Thus, we can rewrite the expression:
\[ \cos^{-1}\left(\cos\left(\frac{\pi}{4}\right)\cos\left(\frac{7\pi}{5}\right) + \sin\left(\frac{\pi}{4}\right)\sin\left(\frac{7\pi}{5}\right)\right) \]
This is the cosine of a sum:
\[ \cos^{-1}\left(\cos\left(\frac{7\pi}{5} - \frac{\pi}{4}\right)\right) \]

### Step 4: Calculate the Angle
Now we need to calculate:
\[ \frac{7\pi}{5} - \frac{\pi}{4} \]
Finding a common denominator (which is 20):
\[ \frac{7\pi}{5} = \frac{28\pi}{20}, \quad \frac{\pi}{4} = \frac{5\pi}{20} \]
Thus:
\[ \frac{7\pi}{5} - \frac{\pi}{4} = \frac{28\pi}{20} - \frac{5\pi}{20} = \frac{23\pi}{20} \]

### Step 5: Evaluating the Inverse Cosine
Now we have:
\[ \cos^{-1}\left(\cos\left(\frac{23\pi}{20}\right)\right) \]
Since \(\frac{23\pi}{20}\) is in the range of \(\cos^{-1}\) which is \([0, \pi]\), we can directly evaluate:
\[ \cos^{-1}\left(\cos\left(\frac{23\pi}{20}\right)\right) = \frac{23\pi}{20} \]

### Step 6: Setting Up the Equation
Now we can set:
\[ \frac{p\pi}{q} = \frac{23\pi}{20} \]
From this, we can see that \(p = 23\) and \(q = 20\).

### Step 7: Finding \(q - p\)
Finally, we need to calculate:
\[ q - p = 20 - 23 = -3 \]

### Final Answer
Thus, the value of \(q - p\) is:
\[ \boxed{-3} \]