\tan ^2\left(x\right)\sec \left(x\right)+\sec ^3\left(x\right)

We are tasked with simplifying the expression:

\[
\tan^2(x) \sec(x) + \sec^3(x)
\]

### Step 1: Express everything in terms of \(\sin(x)\) and \(\cos(x)\)
We know the following trigonometric identities:
- \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)
- \(\sec(x) = \frac{1}{\cos(x)}\)

Thus, we can rewrite the expression as follows:

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

Simplifying both terms:

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

### Step 2: Combine the terms
Since both terms have the same denominator, \(\cos^3(x)\), we can combine them:

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

### Step 3: Use the Pythagorean identity
We apply the Pythagorean identity: \(\sin^2(x) + \cos^2(x) = 1\), which allows us to replace \(\sin^2(x) + 1\) with \(2 - \cos^2(x)\):

\[
\frac{2 - \cos^2(x)}{\cos^3(x)}
\]

### Final expression:
The simplified form of the given expression is:

\[
\frac{2 - \cos^2(x)}{\cos^3(x)}
\]

This is the simplest form of the expression.

Would you like a further breakdown of any part?

Here are five related questions you might find interesting:
1. How do you simplify other trigonometric expressions involving \(\tan\) and \(\sec\)?
2. Can this expression be further evaluated for specific values of \(x\)?
3. What is the derivative of this simplified expression?
4. How would you approach solving an equation that involves this expression?
5. How can we convert this expression into a form involving only sine functions?

**Tip:** Memorizing key trigonometric identities like the Pythagorean identity can help simplify complex trigonometric expressions quickly.

This page offers a detailed step-by-step guide on how to simplify the trigonometric expression tan^2(x) sec(x) + sec^3(x) using trigonometric identities. Suitable for students in Grades 10-12, this problem reinforces concepts like secant, tangent, and the Pythagorean identity.

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