To simplify the given expression \(\frac{\sin^3x + \csc^3x}{\sin^2x + \cot^2x}\), let's proceed step by step.

### Step 1: Express the terms in simpler forms
Recall the following identities:
- \(\csc x = \frac{1}{\sin x}\)
- \(\cot x = \frac{\cos x}{\sin x}\)

Substituting these into the expression:

\[
\frac{\sin^3x + \left(\frac{1}{\sin x}\right)^3}{\sin^2x + \left(\frac{\cos x}{\sin x}\right)^2}
\]

This simplifies to:

\[
\frac{\sin^3x + \frac{1}{\sin^3x}}{\sin^2x + \frac{\cos^2x}{\sin^2x}}
\]

### Step 2: Simplify the denominator
The denominator can be simplified using the Pythagorean identity \(\sin^2x + \cos^2x = 1\):

\[
\sin^2x + \frac{\cos^2x}{\sin^2x} = \frac{\sin^4x + \cos^2x}{\sin^2x} = \frac{\sin^4x + (1 - \sin^2x)}{\sin^2x}
\]

So, the denominator simplifies to:

\[
\frac{1}{\sin^2x}
\]

### Step 3: Substitute back into the expression
Now, substituting the simplified denominator into the expression:

\[
\frac{\sin^3x + \frac{1}{\sin^3x}}{\frac{1}{\sin^2x}}
\]

Simplify by multiplying the numerator by \(\sin^2x\):

\[
\sin^5x + \frac{\sin^2x}{\sin^3x} = \sin^5x + \frac{1}{\sin x} = \sin^5x + \csc x
\]

So, the simplified expression is:

\[
\sin^5x + \csc x
\]

This is the final simplified form.

Would you like further details or have any questions?

### Related Questions:
1. How would the expression change if we used \(\cos x\) instead of \(\sin x\)?
2. Can this method be applied to other trigonometric identities?
3. What happens if we square the original expression?
4. How would you differentiate the simplified expression with respect to \(x\)?
5. What would the simplified expression be if \(x = \frac{\pi}{4}\)?

**Tip:** Simplifying trigonometric expressions often requires a good grasp of fundamental identities like the Pythagorean identity and reciprocal identities.

Learn how to simplify the trigonometric expression sin^3x + csc^3x / sin^2x + cot^2x step-by-step. This problem involves trigonometric identities, including reciprocal and Pythagorean identities. Suitable for advanced high school and college-level students.

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