Convert the sums or differences into products:
`sin 61^(@) - cos 39^(@)`

Answer

To convert the expression \( \sin 61^\circ - \cos 39^\circ \) into a product, we can follow these steps:

### Step 1: Rewrite \(\cos 39^\circ\)
We can use the co-function identity for cosine, which states that:
\[ \cos(90^\circ - \theta) = \sin(\theta) \]
Thus, we can rewrite \(\cos 39^\circ\) as:
\[ \cos 39^\circ = \sin(90^\circ - 39^\circ) = \sin 51^\circ \]

### Step 2: Substitute into the expression
Now we can substitute this back into the original expression:
\[ \sin 61^\circ - \cos 39^\circ = \sin 61^\circ - \sin 51^\circ \]

### Step 3: Apply the sine difference formula
We can use the sine difference formula:
\[ \sin A - \sin B = 2 \cdot \cos\left(\frac{A+B}{2}\right) \cdot \sin\left(\frac{A-B}{2}\right) \]
Here, let \( A = 61^\circ \) and \( B = 51^\circ \).

### Step 4: Calculate \( A + B \) and \( A - B \)
Now, we calculate:
\[ A + B = 61^\circ + 51^\circ = 112^\circ \]
\[ A - B = 61^\circ - 51^\circ = 10^\circ \]

### Step 5: Substitute into the sine difference formula
Now we can substitute these values into the sine difference formula:
\[ \sin 61^\circ - \sin 51^\circ = 2 \cdot \cos\left(\frac{112^\circ}{2}\right) \cdot \sin\left(\frac{10^\circ}{2}\right) \]
This simplifies to:
\[ = 2 \cdot \cos(56^\circ) \cdot \sin(5^\circ) \]

### Final Answer
Thus, the expression \( \sin 61^\circ - \cos 39^\circ \) can be expressed as:
\[ \boxed{2 \cdot \cos(56^\circ) \cdot \sin(5^\circ)} \]

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