The vlaue of `cos^(2)76^(@)+cos^(2)16-cos76^(@)cos16^(@),`is

Options

A. `1//2`
B. 0
C. `-1//4`
D. `3//4`

Answer

To solve the expression \( \cos^2 76^\circ + \cos^2 16^\circ - \cos 76^\circ \cos 16^\circ \), we can follow these steps:

### Step-by-Step Solution

1. **Multiply and Divide by 2**:
\[ \cos^2 76^\circ + \cos^2 16^\circ - \cos 76^\circ \cos 16^\circ = \frac{1}{2}(2\cos^2 76^\circ + 2\cos^2 16^\circ - 2\cos 76^\circ \cos 16^\circ) \]

2. **Use the Identity**:
Recall the identity \( 2\cos^2 \theta = 1 + \cos 2\theta \). Applying this:
\[ 2\cos^2 76^\circ = 1 + \cos 152^\circ \]
\[ 2\cos^2 16^\circ = 1 + \cos 32^\circ \]

Therefore, substituting these into our equation:
\[ \frac{1}{2} \left( (1 + \cos 152^\circ) + (1 + \cos 32^\circ) - 2\cos 76^\circ \cos 16^\circ \right) \]

3. **Combine the Terms**:
Combine the constant terms:
\[ = \frac{1}{2} \left( 2 + \cos 152^\circ + \cos 32^\circ - 2\cos 76^\circ \cos 16^\circ \right) \]

4. **Use the Product-to-Sum Formula**:
Recall the product-to-sum formula:
\[ 2\cos A \cos B = \cos(A+B) + \cos(A-B) \]
Applying this to \( 2\cos 76^\circ \cos 16^\circ \):
\[ 2\cos 76^\circ \cos 16^\circ = \cos(92^\circ) + \cos(60^\circ) \]

5. **Substitute Back**:
Substitute back into the equation:
\[ = \frac{1}{2} \left( 2 + \cos 152^\circ + \cos 32^\circ - (\cos 92^\circ + \cos 60^\circ) \right) \]

6. **Evaluate the Cosine Values**:
We know:
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \cos 92^\circ \) is negative but we will keep it as is for now.

Now, we can simplify:
\[ = \frac{1}{2} \left( 2 + \cos 152^\circ + \cos 32^\circ - \cos 92^\circ - \frac{1}{2} \right) \]

7. **Final Calculation**:
Combine the constants:
\[ = \frac{1}{2} \left( \frac{3}{2} + \cos 152^\circ + \cos 32^\circ - \cos 92^\circ \right) \]

8. **Using Known Values**:
Since \( \cos 152^\circ = -\cos 28^\circ \) and \( \cos 32^\circ = \cos 32^\circ \), we can evaluate the final expression.

After evaluating, we find that:
\[ \text{Final Value} = \frac{3}{4} \]

### Conclusion
Thus, the value of \( \cos^2 76^\circ + \cos^2 16^\circ - \cos 76^\circ \cos 16^\circ \) is \( \frac{3}{4} \).