`cos^(2)""(pi)/(16)+cos^(2)""(3pi)/(16)+cos^(2)""(5pi)/(16)+cos^(2)""(7pi)/(16)=?`

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the expression \( \cos^2\left(\frac{\pi}{16}\right) + \cos^2\left(\frac{3\pi}{16}\right) + \cos^2\left(\frac{5\pi}{16}\right) + \cos^2\left(\frac{7\pi}{16}\right) \), we can use the property of complementary angles in trigonometry.

### Step-by-step Solution:

1. **Identify Complementary Angles:**
- The angles \( \frac{\pi}{16} \) and \( \frac{7\pi}{16} \) are complementary because:
\[ \frac{\pi}{16} + \frac{7\pi}{16} = \frac{8\pi}{16} = \frac{\pi}{2} \]
- Similarly, \( \frac{3\pi}{16} \) and \( \frac{5\pi}{16} \) are complementary:
\[ \frac{3\pi}{16} + \frac{5\pi}{16} = \frac{8\pi}{16} = \frac{\pi}{2} \]

2. **Use the Identity for Complementary Angles:**
- For complementary angles, we know that:
\[ \cos^2(x) + \sin^2(x) = 1 \]
- Therefore, we can rewrite:
\[ \cos^2\left(\frac{7\pi}{16}\right) = \sin^2\left(\frac{\pi}{16}\right) \]
\[ \cos^2\left(\frac{5\pi}{16}\right) = \sin^2\left(\frac{3\pi}{16}\right) \]

3. **Rewrite the Expression:**
- Now substituting these identities into the original expression:
\[ \cos^2\left(\frac{\pi}{16}\right) + \sin^2\left(\frac{\pi}{16}\right) + \cos^2\left(\frac{3\pi}{16}\right) + \sin^2\left(\frac{3\pi}{16}\right) \]

4. **Group the Terms:**
- Group the terms:
\[ \left(\cos^2\left(\frac{\pi}{16}\right) + \sin^2\left(\frac{\pi}{16}\right)\right) + \left(\cos^2\left(\frac{3\pi}{16}\right) + \sin^2\left(\frac{3\pi}{16}\right)\right) \]

5. **Apply the Identity:**
- Using the identity \( \cos^2(x) + \sin^2(x) = 1 \):
\[ 1 + 1 = 2 \]

6. **Final Answer:**
- Therefore, the value of the expression is:
\[ \boxed{2} \]