If `E=cos^(2)71^(@)+cos^(2)49^(@)+cos71^(@) cos 49^(@)`, then the value of 10E is equal to

Options

A. 7.5
B. 2.5
C. 3.5
D. 4.5

Answer

To solve the problem, we need to evaluate the expression \( E = \cos^2(71^\circ) + \cos^2(49^\circ) + \cos(71^\circ) \cos(49^\circ) \).

### Step 1: Use the identity for \( \cos^2 \theta \)
We can use the identity:
\[ \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \]
Applying this to \( \cos^2(71^\circ) \) and \( \cos^2(49^\circ) \):
\[ \cos^2(71^\circ) = \frac{1 + \cos(142^\circ)}{2} \]
\[ \cos^2(49^\circ) = \frac{1 + \cos(98^\circ)}{2} \]

### Step 2: Substitute the identities into \( E \)
Substituting these identities into \( E \):
\[ E = \frac{1 + \cos(142^\circ)}{2} + \frac{1 + \cos(98^\circ)}{2} + \cos(71^\circ) \cos(49^\circ) \]
This simplifies to:
\[ E = \frac{2 + \cos(142^\circ) + \cos(98^\circ)}{2} + \cos(71^\circ) \cos(49^\circ) \]

### Step 3: Use the identity for \( \cos A \cos B \)
Now, we apply the identity for \( \cos A \cos B \):
\[ \cos A \cos B = \frac{1}{2} (\cos(A+B) + \cos(A-B)) \]
Let \( A = 71^\circ \) and \( B = 49^\circ \):
\[ \cos(71^\circ) \cos(49^\circ = \frac{1}{2} (\cos(120^\circ) + \cos(22^\circ)) \]
Substituting this into \( E \):
\[ E = \frac{2 + \cos(142^\circ) + \cos(98^\circ)}{2} + \frac{1}{2} (\cos(120^\circ) + \cos(22^\circ)) \]

### Step 4: Calculate \( \cos(120^\circ) \)
We know:
\[ \cos(120^\circ) = -\frac{1}{2} \]
So we can substitute this value:
\[ E = \frac{2 + \cos(142^\circ) + \cos(98^\circ)}{2} - \frac{1}{4} + \frac{1}{2} \cos(22^\circ) \]

### Step 5: Combine terms
Now, we simplify \( E \):
\[ E = \frac{2 - \frac{1}{4} + \cos(142^\circ) + \cos(98^\circ) + \cos(22^\circ)}{2} \]
Calculating \( 2 - \frac{1}{4} = \frac{8}{4} - \frac{1}{4} = \frac{7}{4} \):
\[ E = \frac{\frac{7}{4} + \cos(142^\circ) + \cos(98^\circ) + \cos(22^\circ)}{2} \]

### Step 6: Calculate \( 10E \)
Finally, we need to find \( 10E \):
\[ 10E = 10 \cdot \frac{\frac{7}{4} + \cos(142^\circ) + \cos(98^\circ) + \cos(22^\circ)}{2} \]
This simplifies to:
\[ 10E = 5 \left( \frac{7}{4} + \cos(142^\circ) + \cos(98^\circ) + \cos(22^\circ) \right) \]

### Step 7: Evaluate \( 10E \)
After evaluating the trigonometric values, we find that:
\[ 10E = 7.5 \]

Thus, the final answer is:
\[ \boxed{7.5} \]