If `(sinalpha)/(sinbeta)=(cosgamma)/(cosdelta), then (sin((alpha-beta)/2)*cos((alpha+beta)/2)*cosdelta)/(sin((delta-gamma)/2)*sin((delta+gamma)/2)*sinbeta)` is equal to

Answer

To solve the problem, we need to simplify the expression

\[ \frac{\sin\left(\frac{\alpha - \beta}{2}\right) \cos\left(\frac{\alpha + \beta}{2}\right) \cos \delta}{\sin\left(\frac{\delta - \gamma}{2}\right) \sin\left(\frac{\delta + \gamma}{2}\right) \sin \beta} \]

given that

\[ \frac{\sin \alpha}{\sin \beta} = \frac{\cos \gamma}{\cos \delta}. \]

### Step 1: Use the identities for sine and cosine

We will use the following identities:

1. \(\sin A \cos B = \frac{1}{2} \left(\sin(A + B) + \sin(A - B)\right)\)
2. \(\sin A \sin B = \frac{1}{2} \left(\cos(A - B) - \cos(A + B)\right)\)

### Step 2: Expand the numerator

Using the first identity on the numerator:

\[ \sin\left(\frac{\alpha - \beta}{2}\right) \cos\left(\frac{\alpha + \beta}{2}\right) = \frac{1}{2} \left(\sin\left(\frac{\alpha - \beta}{2} + \frac{\alpha + \beta}{2}\right) + \sin\left(\frac{\alpha - \beta}{2} - \frac{\alpha + \beta}{2}\right)\right) \]

This simplifies to:

\[ \frac{1}{2} \left(\sin \alpha + \sin(-\beta)\right) = \frac{1}{2} \left(\sin \alpha - \sin \beta\right) \]

So, the numerator becomes:

\[ \frac{1}{2} \left(\sin \alpha - \sin \beta\right) \cos \delta \]

### Step 3: Expand the denominator

Using the second identity on the denominator:

\[ \sin\left(\frac{\delta - \gamma}{2}\right) \sin\left(\frac{\delta + \gamma}{2}\right) = \frac{1}{2} \left(\cos\left(\frac{\delta - \gamma}{2} - \frac{\delta + \gamma}{2}\right) - \cos\left(\frac{\delta - \gamma}{2} + \frac{\delta + \gamma}{2}\right)\right) \]

This simplifies to:

\[ \frac{1}{2} \left(\cos(-\gamma) - \cos \delta\right) = \frac{1}{2} \left(\cos \gamma - \cos \delta\right) \]

Thus, the denominator becomes:

\[ \frac{1}{2} \left(\cos \gamma - \cos \delta\right) \sin \beta \]

### Step 4: Substitute into the expression

Now substituting the expanded numerator and denominator into the original expression, we have:

\[ \frac{\frac{1}{2} \left(\sin \alpha - \sin \beta\right) \cos \delta}{\frac{1}{2} \left(\cos \gamma - \cos \delta\right) \sin \beta} \]

The \(\frac{1}{2}\) cancels out:

\[ \frac{\left(\sin \alpha - \sin \beta\right) \cos \delta}{\left(\cos \gamma - \cos \delta\right) \sin \beta} \]

### Step 5: Use the given condition

From the given condition \(\frac{\sin \alpha}{\sin \beta} = \frac{\cos \gamma}{\cos \delta}\), we can express \(\sin \alpha \cos \delta\) in terms of \(\sin \beta \cos \gamma\):

\[ \sin \alpha \cos \delta = \sin \beta \cos \gamma \]

Substituting this into our expression gives:

\[ \frac{\sin \beta \cos \gamma - \sin \beta \cos \delta}{\left(\cos \gamma - \cos \delta\right) \sin \beta} \]

### Step 6: Simplify the expression

Factoring out \(\sin \beta\) from the numerator:

\[ \frac{\sin \beta (\cos \gamma - \cos \delta)}{(\cos \gamma - \cos \delta) \sin \beta} \]

The \(\sin \beta\) cancels out, and we are left with:

\[ 1 \]

### Final Answer

Thus, the value of the given expression is:

\[ \boxed{1} \]