Find the value of 8 `cos10^(@)cos20^(@)cos40^(@)` .

Options

A. `tan80`
B. `cot10^(@)`
C. `tan80^(@)` or `cot10^(@)`
D. None of these

Answer

To find the value of \( 8 \cos 10^\circ \cos 20^\circ \cos 40^\circ \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x = 8 \cos 10^\circ \cos 20^\circ \cos 40^\circ \]

### Step 2: Factor out constants
We can rewrite \( 8 \) as \( 2^3 \):
\[ x = 2^3 \cos 10^\circ \cos 20^\circ \cos 40^\circ \]

### Step 3: Use the identity for product of cosines
We can use the identity:
\[ 2 \cos A \cos B = \cos(A + B) + \cos(A - B) \]
to combine the cosines. First, we will combine \( \cos 20^\circ \) and \( \cos 40^\circ \).

### Step 4: Combine \( \cos 20^\circ \) and \( \cos 40^\circ \)
Using the identity:
\[ 2 \cos 20^\circ \cos 40^\circ = \cos(20^\circ + 40^\circ) + \cos(20^\circ - 40^\circ) = \cos 60^\circ + \cos (-20^\circ) \]
Since \( \cos(-\theta) = \cos(\theta) \), we have:
\[ 2 \cos 20^\circ \cos 40^\circ = \cos 60^\circ + \cos 20^\circ \]
Now, substituting \( \cos 60^\circ = \frac{1}{2} \):
\[ 2 \cos 20^\circ \cos 40^\circ = \frac{1}{2} + \cos 20^\circ \]

### Step 5: Substitute back into the expression for \( x \)
Now we can substitute this back into our expression for \( x \):
\[ x = 2 \cos 10^\circ \left(\frac{1}{2} + \cos 20^\circ\right) \]
Distributing \( 2 \cos 10^\circ \):
\[ x = \cos 10^\circ + 2 \cos 10^\circ \cos 20^\circ \]

### Step 6: Combine \( \cos 10^\circ \) and \( \cos 20^\circ \)
Now we can apply the identity again to \( 2 \cos 10^\circ \cos 20^\circ \):
\[ 2 \cos 10^\circ \cos 20^\circ = \cos(10^\circ + 20^\circ) + \cos(10^\circ - 20^\circ) = \cos 30^\circ + \cos (-10^\circ) \]
Again, using \( \cos(-\theta) = \cos(\theta) \):
\[ 2 \cos 10^\circ \cos 20^\circ = \cos 30^\circ + \cos 10^\circ \]
Substituting \( \cos 30^\circ = \frac{\sqrt{3}}{2} \):
\[ 2 \cos 10^\circ \cos 20^\circ = \frac{\sqrt{3}}{2} + \cos 10^\circ \]

### Step 7: Substitute back into \( x \)
Now substituting this back into our expression for \( x \):
\[ x = \cos 10^\circ + \left(\frac{\sqrt{3}}{2} + \cos 10^\circ\right) = 2 \cos 10^\circ + \frac{\sqrt{3}}{2} \]

### Step 8: Final value
Thus, the final value of \( x \) is:
\[ x = 2 \cos 10^\circ + \frac{\sqrt{3}}{2} \]