Let `f(x)=(sinx+3sin3x+5sin5x+3sin7x)/(sin2x+2sin4x+3sin6x)`, wherever defined. If `x_(1)+x_(2)=(pi)/(2)`, where `f(x)` is defined at `x_(1) and x_(2)`, then `f^(2)(x_(1))+f^(2)(x_(2))` is

Options

A. `cos^(2)x`
B. `sin^(2)x`
C. `4`
D. `1`

Answer

To solve the problem, we need to evaluate the expression \( f^2(x_1) + f^2(x_2) \) given that \( f(x) = \frac{\sin x + 3\sin 3x + 5\sin 5x + 3\sin 7x}{\sin 2x + 2\sin 4x + 3\sin 6x} \) and \( x_1 + x_2 = \frac{\pi}{2} \).

### Step 1: Simplify \( f(x) \)

We start with the function:

\[ f(x) = \frac{\sin x + 3\sin 3x + 5\sin 5x + 3\sin 7x}{\sin 2x + 2\sin 4x + 3\sin 6x} \]

**Hint:** Look for patterns or identities that can help simplify the numerator and denominator.

### Step 2: Use trigonometric identities

Using the identities for sine, we can rewrite parts of the numerator and denominator.

1. **Numerator:**
- Rewrite \( 5\sin 5x \) as \( 3\sin 5x + 2\sin 5x \)
- Rewrite \( 3\sin 7x \) as \( 2\sin 7x + \sin 7x \)

This gives us:

\[ \sin x + 3\sin 3x + 3\sin 5x + 2\sin 5x + 2\sin 7x + \sin 7x \]

2. **Denominator:**
- Rewrite \( \sin 6x \) as \( 2\sin 4x + \sin 2x \)

This gives us:

\[ \sin 2x + 2\sin 4x + 3\sin 6x = \sin 2x + 2\sin 4x + 3(2\sin 4x + \sin 2x) \]

**Hint:** Try to factor out common terms and apply sine addition formulas where applicable.

### Step 3: Factor and simplify

After applying identities and factoring, we find that:

\[ f(x) = 2\cos x \]

**Hint:** Check if the simplification leads to a form that is easier to evaluate.

### Step 4: Evaluate \( f(x_1) \) and \( f(x_2) \)

Given \( x_1 + x_2 = \frac{\pi}{2} \):

\[ f(x_1) = 2\cos x_1 \]
\[ f(x_2) = 2\cos x_2 = 2\sin x_1 \quad (\text{since } \cos(\frac{\pi}{2} - x) = \sin x) \]

### Step 5: Calculate \( f^2(x_1) + f^2(x_2) \)

Now we compute:

\[ f^2(x_1) + f^2(x_2) = (2\cos x_1)^2 + (2\sin x_1)^2 \]

This simplifies to:

\[ = 4\cos^2 x_1 + 4\sin^2 x_1 \]

### Step 6: Use the Pythagorean identity

Using the identity \( \cos^2 x + \sin^2 x = 1 \):

\[ = 4(\cos^2 x_1 + \sin^2 x_1) = 4 \cdot 1 = 4 \]

### Final Answer

Thus, the value of \( f^2(x_1) + f^2(x_2) \) is:

\[ \boxed{4} \]