State Whether it is true or false.
`d/(dx)(cotx)=sec^(2)x.`

Answer

To determine whether the statement \( \frac{d}{dx}(\cot x) = \sec^2 x \) is true or false, we will differentiate \( \cot x \) and compare the result with \( \sec^2 x \).

### Step-by-Step Solution:

1. **Recall the definition of cotangent**:
\[ \cot x = \frac{\cos x}{\sin x} \]

2. **Use the quotient rule for differentiation**:
The quotient rule states that if \( y = \frac{u}{v} \), then:
\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
Here, \( u = \cos x \) and \( v = \sin x \).

3. **Differentiate \( u \) and \( v \)**:
- \( \frac{du}{dx} = -\sin x \)
- \( \frac{dv}{dx} = \cos x \)

4. **Apply the quotient rule**:
\[ \frac{d}{dx}(\cot x) = \frac{\sin x (-\sin x) - \cos x (\cos x)}{\sin^2 x} \]
Simplifying this gives:
\[ = \frac{-\sin^2 x - \cos^2 x}{\sin^2 x} \]

5. **Use the Pythagorean identity**:
We know that:
\[ \sin^2 x + \cos^2 x = 1 \]
Therefore:
\[ -(\sin^2 x + \cos^2 x) = -1 \]
So we can rewrite the expression as:
\[ \frac{-1}{\sin^2 x} \]

6. **Recognize the final result**:
We know that:
\[ \frac{1}{\sin^2 x} = \csc^2 x \]
Thus:
\[ \frac{d}{dx}(\cot x) = -\csc^2 x \]

7. **Conclusion**:
Since \( \sec^2 x \) is not equal to \( -\csc^2 x \), we conclude that:
\[ \frac{d}{dx}(\cot x) \neq \sec^2 x \]
Therefore, the statement is **false**.