The value of `cos 52^(@) + cos 68^(@) + cos 172^(@)` is

Options

A. 0
B. 1
C. 2
D. `(1)/(sqrt(2))`

Answer

To find the value of \( \cos 52^\circ + \cos 68^\circ + \cos 172^\circ \), we can use the cosine addition formula. Here’s a step-by-step solution:

### Step 1: Group the cosines
We can group \( \cos 52^\circ \) and \( \cos 68^\circ \) together and leave \( \cos 172^\circ \) as is.

### Step 2: Apply the cosine addition formula
The formula for the sum of cosines is:
\[ \cos c + \cos d = 2 \cos\left(\frac{c + d}{2}\right) \cos\left(\frac{c - d}{2}\right) \]
Let \( c = 52^\circ \) and \( d = 68^\circ \).

### Step 3: Calculate \( c + d \) and \( c - d \)
\[ c + d = 52^\circ + 68^\circ = 120^\circ \]
\[ c - d = 52^\circ - 68^\circ = -16^\circ \]

### Step 4: Substitute into the formula
Now we substitute into the formula:
\[ \cos 52^\circ + \cos 68^\circ = 2 \cos\left(\frac{120^\circ}{2}\right) \cos\left(\frac{-16^\circ}{2}\right) \]
This simplifies to:
\[ \cos 52^\circ + \cos 68^\circ = 2 \cos(60^\circ) \cos(-8^\circ) \]

### Step 5: Evaluate \( \cos(60^\circ) \)
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]
Thus, we have:
\[ \cos 52^\circ + \cos 68^\circ = 2 \cdot \frac{1}{2} \cdot \cos(-8^\circ) = \cos(-8^\circ) \]

### Step 6: Use the property of cosine
Using the property \( \cos(-\theta) = \cos(\theta) \):
\[ \cos(-8^\circ) = \cos(8^\circ) \]

### Step 7: Combine with \( \cos 172^\circ \)
Now we add \( \cos 172^\circ \):
\[ \cos 52^\circ + \cos 68^\circ + \cos 172^\circ = \cos(8^\circ) + \cos(172^\circ) \]

### Step 8: Apply the cosine addition formula again
Now we apply the cosine addition formula again:
Let \( e = 8^\circ \) and \( f = 172^\circ \):
\[ \cos 8^\circ + \cos 172^\circ = 2 \cos\left(\frac{8^\circ + 172^\circ}{2}\right) \cos\left(\frac{8^\circ - 172^\circ}{2}\right) \]
Calculating \( e + f \) and \( e - f \):
\[ e + f = 180^\circ \]
\[ e - f = -164^\circ \]

### Step 9: Substitute into the formula
Substituting into the formula gives:
\[ \cos 8^\circ + \cos 172^\circ = 2 \cos(90^\circ) \cos(-82^\circ) \]

### Step 10: Evaluate \( \cos(90^\circ) \)
Since \( \cos(90^\circ) = 0 \):
\[ \cos 8^\circ + \cos 172^\circ = 2 \cdot 0 \cdot \cos(-82^\circ) = 0 \]

### Final Answer
Thus, the value of \( \cos 52^\circ + \cos 68^\circ + \cos 172^\circ \) is:
\[ \boxed{0} \]