`f(x)=cot^(2)x*cos^(2)x, g(x)=cot^(2)x-cos^(2)x`

Answer

To determine if the functions \( f(x) = \cot^2 x \cdot \cos^2 x \) and \( g(x) = \cot^2 x - \cos^2 x \) are identical, we will follow the steps outlined in the video transcript:

### Step 1: Determine the Domain of \( f(x) \)

1. **Identify the components of \( f(x) \)**:
- \( f(x) = \cot^2 x \cdot \cos^2 x \)
- The function \( \cot^2 x \) is defined for all \( x \) except where \( \cos x = 0 \), which occurs at \( x = n\pi + \frac{\pi}{2} \) for \( n \in \mathbb{Z} \).
- The function \( \cos^2 x \) is defined for all \( x \).

2. **Find the domain**:
- The domain of \( f(x) \) is the intersection of the domains of \( \cot^2 x \) and \( \cos^2 x \).
- Thus, the domain of \( f(x) \) is all real numbers except \( x = n\pi + \frac{\pi}{2} \).

### Step 2: Determine the Domain of \( g(x) \)

1. **Identify the components of \( g(x) \)**:
- \( g(x) = \cot^2 x - \cos^2 x \)
- Again, \( \cot^2 x \) is defined for all \( x \) except where \( \cos x = 0 \), and \( \cos^2 x \) is defined for all \( x \).

2. **Find the domain**:
- The domain of \( g(x) \) is also the intersection of the domains of \( \cot^2 x \) and \( \cos^2 x \).
- Thus, the domain of \( g(x) \) is all real numbers except \( x = n\pi + \frac{\pi}{2} \).

### Step 3: Compare the Domains

- Since both \( f(x) \) and \( g(x) \) have the same domain, we can proceed to the next step.

### Step 4: Simplify \( g(x) \)

1. **Rewrite \( g(x) \)**:
- Start with \( g(x) = \cot^2 x - \cos^2 x \).
- Recall that \( \cot x = \frac{\cos x}{\sin x} \), so \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \).

2. **Substitute and simplify**:
- \( g(x) = \frac{\cos^2 x}{\sin^2 x} - \cos^2 x \)
- Factor out \( \cos^2 x \):
\[ g(x) = \cos^2 x \left( \frac{1}{\sin^2 x} - 1 \right) \]
- Simplifying further:
\[ g(x) = \cos^2 x \left( \frac{1 - \sin^2 x}{\sin^2 x} \right) \]
- Since \( 1 - \sin^2 x = \cos^2 x \):
\[ g(x) = \cos^2 x \cdot \frac{\cos^2 x}{\sin^2 x} = \cot^2 x \cdot \cos^2 x \]

### Step 5: Compare \( f(x) \) and \( g(x) \)

- We find that:
\[ g(x) = \cot^2 x \cdot \cos^2 x = f(x) \]
- Therefore, \( f(x) = g(x) \) for all \( x \) in their common domain.

### Conclusion

Since both functions have the same domain and \( f(x) = g(x) \) for all \( x \) in that domain, we conclude that the functions \( f(x) \) and \( g(x) \) are identical.

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