If `f (x)=sin^2 x+ sin^2 (x + (pi)/(3) ) + cos x cos(x+ pi/3)` and g (5/4) = 1, then (gof) x is

Options

A. polynomial of first degree in sin x,cos X
B. a constant function
C. polynomial of 2nd degree in sin x,cos x
D. none

Answer

To solve the problem step by step, we will start with the function \( f(x) \) and simplify it, then we will use the given information about \( g \) to find \( (g \circ f)(x) \).

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

Given:
\[ f(x) = \sin^2 x + \sin^2\left(x + \frac{\pi}{3}\right) + \cos x \cos\left(x + \frac{\pi}{3}\right) \]

We will simplify each term in \( f(x) \).

1. **First Term**: \( \sin^2 x \) remains as is.

2. **Second Term**:
\[ \sin^2\left(x + \frac{\pi}{3}\right) = \left(\sin x \cos\frac{\pi}{3} + \cos x \sin\frac{\pi}{3}\right)^2 \]
Using \( \cos\frac{\pi}{3} = \frac{1}{2} \) and \( \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} \):
\[ = \left(\sin x \cdot \frac{1}{2} + \cos x \cdot \frac{\sqrt{3}}{2}\right)^2 \]
Expanding this:
\[ = \frac{1}{4} \sin^2 x + \frac{3}{4} \cos^2 x + \frac{\sqrt{3}}{2} \sin x \cos x \]

3. **Third Term**:
\[ \cos x \cos\left(x + \frac{\pi}{3}\right) = \cos x \left(\cos x \cos\frac{\pi}{3} - \sin x \sin\frac{\pi}{3}\right) \]
Expanding this:
\[ = \cos x \left(\cos x \cdot \frac{1}{2} - \sin x \cdot \frac{\sqrt{3}}{2}\right) = \frac{1}{2} \cos^2 x - \frac{\sqrt{3}}{2} \sin x \cos x \]

Now, combining all these terms:
\[ f(x) = \sin^2 x + \left(\frac{1}{4} \sin^2 x + \frac{3}{4} \cos^2 x + \frac{\sqrt{3}}{2} \sin x \cos x\right) + \left(\frac{1}{2} \cos^2 x - \frac{\sqrt{3}}{2} \sin x \cos x\right) \]

### Step 2: Combine Like Terms

Combining all the terms:
- The \( \sin^2 x \) terms:
\[ \sin^2 x + \frac{1}{4} \sin^2 x = \frac{5}{4} \sin^2 x \]

- The \( \cos^2 x \) terms:
\[ \frac{3}{4} \cos^2 x + \frac{1}{2} \cos^2 x = \frac{3}{4} \cos^2 x + \frac{2}{4} \cos^2 x = \frac{5}{4} \cos^2 x \]

- The mixed terms:
\[ \frac{\sqrt{3}}{2} \sin x \cos x - \frac{\sqrt{3}}{2} \sin x \cos x = 0 \]

Thus, we have:
\[ f(x) = \frac{5}{4}(\sin^2 x + \cos^2 x) = \frac{5}{4} \]

### Step 3: Finding \( g(f(x)) \)

Since we found that \( f(x) = \frac{5}{4} \), we can substitute this into \( g \):
\[ g(f(x)) = g\left(\frac{5}{4}\right) \]

Given that \( g\left(\frac{5}{4}\right) = 1 \), we conclude:
\[ g(f(x)) = 1 \]

### Final Answer

Thus, \( (g \circ f)(x) = 1 \).

---