Using the identity
`sin^(4) x=3/8-1/2 cos 2x+1/8 cos 4x` or otherwise, if the value of `sin^(4)(pi/7)+sin^(4)((3pi)/(7))+sin^(4)((5pi)/(7))=a/b`, where a and b are coprime, find the value of `(a-b)`.

Answer

To solve the problem, we will use the given identity:

\[ \sin^4 x = \frac{3}{8} - \frac{1}{2} \cos 2x + \frac{1}{8} \cos 4x \]

We need to find the value of:

\[ \sin^4\left(\frac{\pi}{7}\right) + \sin^4\left(\frac{3\pi}{7}\right) + \sin^4\left(\frac{5\pi}{7}\right) \]

### Step 1: Apply the identity for each term

Let \( x = \frac{\pi}{7} \), then we have:

\[ \sin^4\left(\frac{\pi}{7}\right) = \frac{3}{8} - \frac{1}{2} \cos\left(\frac{2\pi}{7}\right) + \frac{1}{8} \cos\left(\frac{4\pi}{7}\right) \]

Next, let \( x = \frac{3\pi}{7} \):

\[ \sin^4\left(\frac{3\pi}{7}\right) = \frac{3}{8} - \frac{1}{2} \cos\left(\frac{6\pi}{7}\right) + \frac{1}{8} \cos\left(\frac{12\pi}{7}\right) \]

And for \( x = \frac{5\pi}{7} \):

\[ \sin^4\left(\frac{5\pi}{7}\right) = \frac{3}{8} - \frac{1}{2} \cos\left(\frac{10\pi}{7}\right) + \frac{1}{8} \cos\left(\frac{20\pi}{7}\right) \]

### Step 2: Combine the results

Now we can sum these three equations:

\[ \sin^4\left(\frac{\pi}{7}\right) + \sin^4\left(\frac{3\pi}{7}\right) + \sin^4\left(\frac{5\pi}{7}\right) = 3 \cdot \frac{3}{8} - \frac{1}{2} \left( \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) + \cos\left(\frac{10\pi}{7}\right) \right) + \frac{1}{8} \left( \cos\left(\frac{4\pi}{7}\right) + \cos\left(\frac{12\pi}{7}\right) + \cos\left(\frac{20\pi}{7}\right) \right) \]

### Step 3: Simplify the cosine terms

Using the property of cosine, we know:

\[ \cos\left(\frac{6\pi}{7}\right) = -\cos\left(\frac{\pi}{7}\right), \quad \cos\left(\frac{10\pi}{7}\right) = -\cos\left(\frac{3\pi}{7}\right), \quad \cos\left(\frac{12\pi}{7}\right) = -\cos\left(\frac{5\pi}{7}\right) \]

Thus, we can simplify:

\[ \cos\left(\frac{2\pi}{7}\right) + \cos\left(\frac{6\pi}{7}\right) + \cos\left(\frac{10\pi}{7}\right) = \cos\left(\frac{2\pi}{7}\right) - \cos\left(\frac{\pi}{7}\right) - \cos\left(\frac{3\pi}{7}\right) \]

And similarly for the other cosine terms.

### Step 4: Calculate the final sum

After substituting these values back into the equation, we can find a common denominator and combine the fractions.

The final result will yield:

\[ \sin^4\left(\frac{\pi}{7}\right) + \sin^4\left(\frac{3\pi}{7}\right) + \sin^4\left(\frac{5\pi}{7}\right) = \frac{21}{16} \]

### Step 5: Identify \( a \) and \( b \)

Here, \( a = 21 \) and \( b = 16 \). Since \( a \) and \( b \) are coprime, we can now find \( a - b \):

\[ a - b = 21 - 16 = 5 \]

### Final Answer

Thus, the value of \( a - b \) is:

\[ \boxed{5} \]