Convert the products into sum or difference. If angles are given in degrees, evaluate from tables.
`2 sin 54 ^(@) sin 66^(@)`

Answer

To solve the problem \( 2 \sin 54^\circ \sin 66^\circ \) and convert it into a sum or difference, we can use the product-to-sum identities in trigonometry. Here’s the step-by-step solution:

### Step 1: Identify the formula
We will use the identity:
\[ 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \]
where \( A = 54^\circ \) and \( B = 66^\circ \).

### Step 2: Substitute the values into the formula
Substituting \( A \) and \( B \) into the formula:
\[ 2 \sin 54^\circ \sin 66^\circ = \cos(54^\circ - 66^\circ) - \cos(54^\circ + 66^\circ) \]

### Step 3: Calculate the angles
Now, calculate \( 54^\circ - 66^\circ \) and \( 54^\circ + 66^\circ \):
\[ 54^\circ - 66^\circ = -12^\circ \]
\[ 54^\circ + 66^\circ = 120^\circ \]

### Step 4: Rewrite the expression
Now we can rewrite the expression:
\[ 2 \sin 54^\circ \sin 66^\circ = \cos(-12^\circ) - \cos(120^\circ) \]

### Step 5: Use the property of cosine
Using the property that \( \cos(-\theta) = \cos(\theta) \):
\[ \cos(-12^\circ) = \cos(12^\circ) \]
Thus, we have:
\[ 2 \sin 54^\circ \sin 66^\circ = \cos(12^\circ) - \cos(120^\circ) \]

### Step 6: Evaluate the cosines
Now, we need to find the values of \( \cos(12^\circ) \) and \( \cos(120^\circ) \):
- From trigonometric tables or calculators:
- \( \cos(12^\circ) \approx 0.978 \)
- \( \cos(120^\circ) = -\frac{1}{2} \)

### Step 7: Substitute the values
Substituting these values back into the expression:
\[ 2 \sin 54^\circ \sin 66^\circ = 0.978 - (-0.5) \]
\[ = 0.978 + 0.5 = 1.478 \]

### Final Answer
Thus, the final value of \( 2 \sin 54^\circ \sin 66^\circ \) is approximately \( 1.478 \).

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