If `f(x)=sin^2x+sin^2(x+pi/3)+cosxcos(x+pi/3)a n dg(5/4)=1,` then `(gof)(x)` is ____________

Explanation

To solve the problem, we need to find \( (g \circ f)(x) \) given that \( f(x) = \sin^2 x + \sin^2\left(x + \frac{\pi}{3}\right) + \cos x \cos\left(x + \frac{\pi}{3}\right) \) and \( g\left(\frac{5}{4}\right) = 1 \).

### Step-by-Step Solution:

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

2. **Use the angle addition formulas**:
- We know that:
\[ \sin\left(x + \frac{\pi}{3}\right) = \sin x \cos\frac{\pi}{3} + \cos x \sin\frac{\pi}{3} = \sin x \cdot \frac{1}{2} + \cos x \cdot \frac{\sqrt{3}}{2} \]
- Therefore,
\[ \sin^2\left(x + \frac{\pi}{3}\right) = \left(\frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x\right)^2 \]
- Expanding this gives:
\[ = \frac{1}{4} \sin^2 x + \frac{\sqrt{3}}{2} \sin x \cos x + \frac{3}{4} \cos^2 x \]

3. **Calculate \( \cos\left(x + \frac{\pi}{3}\right) \)**:
- Using the cosine addition formula:
\[ \cos\left(x + \frac{\pi}{3}\right) = \cos x \cos\frac{\pi}{3} - \sin x \sin\frac{\pi}{3} = \cos x \cdot \frac{1}{2} - \sin x \cdot \frac{\sqrt{3}}{2} \]
- Thus,
\[ \cos x \cos\left(x + \frac{\pi}{3}\right) = \cos x \left(\frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x\right) = \frac{1}{2} \cos^2 x - \frac{\sqrt{3}}{2} \sin x \cos x \]

4. **Combine all parts**:
- Now substituting back into \( f(x) \):
\[ f(x) = \sin^2 x + \left(\frac{1}{4} \sin^2 x + \frac{\sqrt{3}}{2} \sin x \cos x + \frac{3}{4} \cos^2 x\right) + \left(\frac{1}{2} \cos^2 x - \frac{\sqrt{3}}{2} \sin x \cos x\right) \]
- Combine like terms:
\[ = \sin^2 x + \frac{1}{4} \sin^2 x + \frac{3}{4} \cos^2 x + \frac{1}{2} \cos^2 x + \left(\frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2}\right) \sin x \cos x \]
- This simplifies to:
\[ = \frac{5}{4} \sin^2 x + \frac{5}{4} \cos^2 x = \frac{5}{4}(\sin^2 x + \cos^2 x) = \frac{5}{4} \]

5. **Find \( g(f(x)) \)**:
- Since we have 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:
\[ (g \circ f)(x) = 1 \]

### Final Answer:
\[ (g \circ f)(x) = 1 \]