(Secx cscx) /csc^2 - sec^2    find derivative

To find the derivative of the given expression \(\frac{\sec(x) \csc(x)}{\csc^2(x)} - \sec^2(x)\), let's break it down step by step.

### Step 1: Simplify the expression
Start by simplifying the first term:

\[
\frac{\sec(x) \csc(x)}{\csc^2(x)} = \frac{\sec(x)}{\csc(x)}
\]

So, the expression becomes:

\[
\frac{\sec(x)}{\csc(x)} - \sec^2(x)
\]

Now, recall that:

\[
\sec(x) = \frac{1}{\cos(x)}, \quad \csc(x) = \frac{1}{\sin(x)}
\]

Thus:

\[
\frac{\sec(x)}{\csc(x)} = \frac{\frac{1}{\cos(x)}}{\frac{1}{\sin(x)}} = \frac{\sin(x)}{\cos(x)} = \tan(x)
\]

So the simplified expression is:

\[
\tan(x) - \sec^2(x)
\]

### Step 2: Find the derivative
Now, let's differentiate the expression with respect to \(x\):

\[
\frac{d}{dx} \left[\tan(x) - \sec^2(x)\right]
\]

The derivative of \(\tan(x)\) is \(\sec^2(x)\), and the derivative of \(-\sec^2(x)\) is \(-2\sec^2(x) \tan(x)\) (using the chain rule).

So, the derivative is:

\[
\sec^2(x) - 2\sec^2(x)\tan(x)
\]

This is the derivative of the original expression.

Would you like more details on any of these steps? Here are some related questions:

1. How do you differentiate \(\tan(x)\) and \(\sec(x)\)?
2. Can you simplify the expression further?
3. What is the chain rule in differentiation?
4. How does trigonometric identity play a role in differentiation?
5. Can we find the derivative using an alternative method?

**Tip:** Always simplify the expression as much as possible before differentiating; this often makes the process easier.

(Secx cscx) /csc^2 - sec^2 find derivative

Step-by-step solution to finding the derivative of the trigonometric expression (sec(x) csc(x)) / csc^2(x) - sec^2(x). Learn about differentiation using trigonometric identities, simplification, and applying the chain rule. Suitable for advanced high school and college-level calculus students.

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