`cos 60^(@) cos 36^(@) cos 42^(@) cos 78^(@) ` is

Options

A. `1//8`
B. 1
C. `1//16`
D. none of these

Answer

To solve the expression \( \cos 60^\circ \cos 36^\circ \cos 42^\circ \cos 78^\circ \), we will follow these steps:

### Step 1: Evaluate \( \cos 60^\circ \)
The value of \( \cos 60^\circ \) is known to be:
\[ \cos 60^\circ = \frac{1}{2} \]

### Step 2: Evaluate \( \cos 36^\circ \)
The value of \( \cos 36^\circ \) is:
\[ \cos 36^\circ = \frac{\sqrt{5} + 1}{4} \]

### Step 3: Combine \( \cos 42^\circ \) and \( \cos 78^\circ \)
Next, we need to evaluate \( \cos 42^\circ \) and \( \cos 78^\circ \). We can use the identity:
\[ 2 \cos A \cos B = \cos(A + B) + \cos(A - B) \]
Here, let \( A = 42^\circ \) and \( B = 78^\circ \):
\[ 2 \cos 42^\circ \cos 78^\circ = \cos(42^\circ + 78^\circ) + \cos(42^\circ - 78^\circ) \]
Calculating the angles:
\[ 42^\circ + 78^\circ = 120^\circ \quad \text{and} \quad 42^\circ - 78^\circ = -36^\circ \]
Thus, we have:
\[ 2 \cos 42^\circ \cos 78^\circ = \cos 120^\circ + \cos(-36^\circ) \]
Since \( \cos(-x) = \cos(x) \):
\[ \cos(-36^\circ) = \cos 36^\circ \]
And we know:
\[ \cos 120^\circ = -\frac{1}{2} \]
So we can write:
\[ 2 \cos 42^\circ \cos 78^\circ = -\frac{1}{2} + \cos 36^\circ \]

### Step 4: Substitute \( \cos 36^\circ \)
Now substituting \( \cos 36^\circ \):
\[ 2 \cos 42^\circ \cos 78^\circ = -\frac{1}{2} + \frac{\sqrt{5} + 1}{4} \]
To combine these, we need a common denominator:
\[ -\frac{1}{2} = -\frac{2}{4} \]
Thus:
\[ 2 \cos 42^\circ \cos 78^\circ = -\frac{2}{4} + \frac{\sqrt{5} + 1}{4} = \frac{\sqrt{5} - 1}{4} \]
Dividing both sides by 2 gives:
\[ \cos 42^\circ \cos 78^\circ = \frac{\sqrt{5} - 1}{8} \]

### Step 5: Combine all parts
Now we can combine all parts:
\[ \cos 60^\circ \cos 36^\circ \cos 42^\circ \cos 78^\circ = \frac{1}{2} \cdot \frac{\sqrt{5} + 1}{4} \cdot \frac{\sqrt{5} - 1}{8} \]
Calculating this gives:
\[ = \frac{1}{2} \cdot \frac{\sqrt{5}^2 - 1^2}{32} = \frac{1}{2} \cdot \frac{5 - 1}{32} = \frac{1}{2} \cdot \frac{4}{32} = \frac{1}{2} \cdot \frac{1}{8} = \frac{1}{16} \]

### Final Answer
Thus, the value of \( \cos 60^\circ \cos 36^\circ \cos 42^\circ \cos 78^\circ \) is:
\[ \frac{1}{16} \]

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