The value of `2(cos^(2)20^(@)+cos^(2)140^(@)+cos^(2)100^(@))` is

Options

A. `3//2`
B. 3
C. 4
D. none of these

Explanation

To solve the expression \(2(\cos^2 20^\circ + \cos^2 140^\circ + \cos^2 100^\circ)\), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ 2(\cos^2 20^\circ + \cos^2 140^\circ + \cos^2 100^\circ) \]

### Step 2: Use the Identity for Cosine Squared
Recall the identity:
\[ \cos^2 a = \frac{1 + \cos 2a}{2} \]
We can apply this identity to each cosine squared term:
\[ \cos^2 20^\circ = \frac{1 + \cos 40^\circ}{2}, \quad \cos^2 140^\circ = \frac{1 + \cos 280^\circ}{2}, \quad \cos^2 100^\circ = \frac{1 + \cos 200^\circ}{2} \]

### Step 3: Substitute the Identities
Substituting these into the expression gives:
\[ 2\left(\frac{1 + \cos 40^\circ}{2} + \frac{1 + \cos 280^\circ}{2} + \frac{1 + \cos 200^\circ}{2}\right) \]
This simplifies to:
\[ 2\left(\frac{3 + \cos 40^\circ + \cos 280^\circ + \cos 200^\circ}{2}\right) \]

### Step 4: Simplify the Expression
The factor of 2 cancels out:
\[ 3 + \cos 40^\circ + \cos 280^\circ + \cos 200^\circ \]

### Step 5: Evaluate Cosine Terms
Now, we need to evaluate \( \cos 280^\circ \) and \( \cos 200^\circ \):
- \( \cos 280^\circ = \cos(360^\circ - 80^\circ) = \cos 80^\circ \)
- \( \cos 200^\circ = -\cos 20^\circ \) (since \(200^\circ\) is in the third quadrant)

Thus, we can rewrite the expression as:
\[ 3 + \cos 40^\circ + \cos 80^\circ - \cos 20^\circ \]

### Step 6: Use Cosine Addition Identity
Using the cosine addition formula:
\[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
we can combine \( \cos 40^\circ + \cos 80^\circ \):
\[ \cos 40^\circ + \cos 80^\circ = 2 \cos\left(\frac{40^\circ + 80^\circ}{2}\right) \cos\left(\frac{40^\circ - 80^\circ}{2}\right) = 2 \cos 60^\circ \cos (-20^\circ) = \cos 20^\circ \]

### Step 7: Substitute Back
Substituting this back into our expression gives:
\[ 3 + \cos 20^\circ - \cos 20^\circ = 3 \]

### Final Answer
Thus, the value of the original expression is:
\[ \boxed{3} \]