`sin 78^(@) - sin 66^(@) - sin 42^(@) + sin 6^(@) =`

Options

A. `-1`
B. `-1//2`
C. `1//2`
D. 1

Answer

To solve the expression \( \sin 78^\circ - \sin 66^\circ - \sin 42^\circ + \sin 6^\circ \), we will follow these steps:

### Step 1: Rearranging the Expression
We can rearrange the expression as follows:
\[ \sin 78^\circ - \sin 42^\circ - \sin 66^\circ + \sin 6^\circ \]

### Step 2: Grouping Terms
Next, we will group the terms into pairs:
\[ (\sin 78^\circ - \sin 42^\circ) + (\sin 6^\circ - \sin 66^\circ) \]

### Step 3: Applying the Sine Difference Formula
We will use the sine difference formula:
\[ \sin A - \sin B = 2 \sin\left(\frac{A - B}{2}\right) \cos\left(\frac{A + B}{2}\right) \]

#### For the first pair:
Let \( A = 78^\circ \) and \( B = 42^\circ \):
\[ \sin 78^\circ - \sin 42^\circ = 2 \sin\left(\frac{78^\circ - 42^\circ}{2}\right) \cos\left(\frac{78^\circ + 42^\circ}{2}\right) \]
Calculating:
\[ = 2 \sin\left(\frac{36^\circ}{2}\right) \cos\left(\frac{120^\circ}{2}\right) = 2 \sin(18^\circ) \cos(60^\circ) \]

#### For the second pair:
Let \( A = 6^\circ \) and \( B = 66^\circ \):
\[ \sin 6^\circ - \sin 66^\circ = 2 \sin\left(\frac{6^\circ - 66^\circ}{2}\right) \cos\left(\frac{6^\circ + 66^\circ}{2}\right) \]
Calculating:
\[ = 2 \sin\left(\frac{-60^\circ}{2}\right) \cos\left(\frac{72^\circ}{2}\right) = 2 \sin(-30^\circ) \cos(36^\circ) \]

### Step 4: Substituting Known Values
Now we substitute the known values:
\[ \cos(60^\circ) = \frac{1}{2}, \quad \sin(18^\circ) = \frac{\sqrt{5} - 1}{4}, \quad \sin(-30^\circ) = -\frac{1}{2}, \quad \cos(36^\circ) = \frac{\sqrt{5} + 1}{4} \]

### Step 5: Putting It All Together
Now substituting back into the expression:
\[ 2 \sin(18^\circ) \cos(60^\circ) + 2 \sin(-30^\circ) \cos(36^\circ) \]
This simplifies to:
\[ 2 \cdot \frac{\sqrt{5} - 1}{4} \cdot \frac{1}{2} + 2 \cdot \left(-\frac{1}{2}\right) \cdot \frac{\sqrt{5} + 1}{4} \]
Calculating each part:
\[ = \frac{\sqrt{5} - 1}{4} - \frac{\sqrt{5} + 1}{4} \]
Combining the fractions:
\[ = \frac{(\sqrt{5} - 1) - (\sqrt{5} + 1)}{4} = \frac{\sqrt{5} - 1 - \sqrt{5} - 1}{4} = \frac{-2}{4} = -\frac{1}{2} \]

### Final Answer
Thus, the final answer is:
\[ \boxed{-\frac{1}{2}} \]