If `f(x)=cos^(2)x+cos^(2)(x+(pi)/(3))+sinxsin(x+(pi)/(3))` and `g((5)/(4))=3`, then `(d)/(dx)(gof(x))`=

Options

A. 1
B. 0
C. -1
D. none of these

Answer

To solve the problem step by step, we need to differentiate the composition of functions \( g(f(x)) \) where \( f(x) \) is given as:

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

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

We start by simplifying \( f(x) \).

1. **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,
\[ \cos^2\left(x + \frac{\pi}{3}\right) = \left(\frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x\right)^2 \]
Expanding this:
\[ = \frac{1}{4} \cos^2 x - \frac{\sqrt{3}}{2} \cos x \sin x + \frac{3}{4} \sin^2 x \]

2. **Using the sine addition formula**:
\[ \sin\left(x + \frac{\pi}{3}\right) = \sin x \cos\frac{\pi}{3} + \cos x \sin\frac{\pi}{3} = \sin x \cdot \frac{1}{2} + \cos x \cdot \frac{\sqrt{3}}{2} \]
Therefore,
\[ \sin x \sin\left(x + \frac{\pi}{3}\right) = \sin x \left(\frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x\right) \]
Expanding this:
\[ = \frac{1}{2} \sin^2 x + \frac{\sqrt{3}}{2} \sin x \cos x \]

Combining these results, we have:
\[ f(x) = \cos^2 x + \left(\frac{1}{4} \cos^2 x - \frac{\sqrt{3}}{2} \cos x \sin x + \frac{3}{4} \sin^2 x\right) + \left(\frac{1}{2} \sin^2 x + \frac{\sqrt{3}}{2} \sin x \cos x\right) \]

### Step 2: Combine like terms

Combining all terms:
\[ f(x) = \cos^2 x + \frac{1}{4} \cos^2 x + \frac{3}{4} \sin^2 x + \frac{1}{2} \sin^2 x + \left(-\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2}\right) \cos x \sin x \]
This simplifies to:
\[ f(x) = \frac{5}{4} \cos^2 x + \frac{5}{4} \sin^2 x = \frac{5}{4}(\cos^2 x + \sin^2 x) = \frac{5}{4} \]

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

Since \( f(x) = \frac{5}{4} \) is a constant function, we find the derivative:
\[ \frac{d}{dx} g(f(x)) = g'(f(x)) \cdot f'(x) \]

### Step 4: Calculate \( f'(x) \)

Since \( f(x) \) is a constant:
\[ f'(x) = 0 \]

### Step 5: Final result

Thus, we have:
\[ \frac{d}{dx} g(f(x)) = g'\left(\frac{5}{4}\right) \cdot 0 = 0 \]

### Conclusion

The final answer is:
\[ \frac{d}{dx} g(f(x)) = 0 \]

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