Can you solve this without using partial fraction but using trig sub instead?  ((sec^2(x) +1)tan^2(x) ) /(tan^3(x) + 1) 

Sure! We can use trigonometric identities and substitutions to simplify and solve the expression:

Given expression:
\[
\frac{(\sec^2(x) + 1) \tan^2(x)}{\tan^3(x) + 1}
\]

### Step-by-Step Solution

#### Step 1: Use trigonometric identities
We start by recalling the Pythagorean identity for secant and tangent:
\[
\sec^2(x) = 1 + \tan^2(x)
\]

Substitute this identity into the given expression:
\[
\frac{((1 + \tan^2(x)) + 1)\tan^2(x)}{\tan^3(x) + 1}
\]

Simplifying the numerator:
\[
1 + \tan^2(x) + 1 = 2 + \tan^2(x)
\]
Thus, the expression becomes:
\[
\frac{(2 + \tan^2(x)) \tan^2(x)}{\tan^3(x) + 1}
\]

#### Step 2: Simplify the expression
Distribute \(\tan^2(x)\) in the numerator:
\[
\frac{2\tan^2(x) + \tan^4(x)}{\tan^3(x) + 1}
\]

Now, observe that both the numerator and denominator involve powers of \(\tan(x)\). This suggests that we may benefit from a substitution.

#### Step 3: Use substitution
Let \( t = \tan(x) \), so we rewrite the expression in terms of \(t\):
\[
\frac{2t^2 + t^4}{t^3 + 1}
\]

#### Step 4: Factor the numerator
Notice that the numerator can be factored as follows:
\[
2t^2 + t^4 = t^2(t^2 + 2)
\]
Thus, the expression becomes:
\[
\frac{t^2(t^2 + 2)}{t^3 + 1}
\]

#### Step 5: Simplify
The denominator \(t^3 + 1\) is a sum of cubes, which can be factored using the identity:
\[
t^3 + 1 = (t + 1)(t^2 - t + 1)
\]

So the expression becomes:
\[
\frac{t^2(t^2 + 2)}{(t + 1)(t^2 - t + 1)}
\]

At this point, the expression is fully simplified. No further simplification or trigonometric substitution is required.

### Final Answer:
\[
\frac{t^2(t^2 + 2)}{(t + 1)(t^2 - t + 1)}
\]
where \(t = \tan(x)\).

Would you like more details on any step or have additional questions?

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Here are some related questions:
1. How do you factor the sum of cubes formula?
2. What is the Pythagorean identity in trigonometry?
3. How does substitution help in solving trigonometric expressions?
4. Can you explain how to factor trinomials?
5. Why does the tangent function appear frequently in calculus problems?

**Tip:** Substitution is a powerful tool for simplifying expressions, especially when working with trigonometric functions.

Can you solve this without using partial fraction but using trig sub instead? ((sec^2(x) +1)tan^2(x)) /(tan^3(x) + 1)

Learn how to solve the trigonometric expression ((sec^2(x) + 1)tan^2(x)) / (tan^3(x) + 1) using trigonometric substitution and factorization. This solution covers key concepts like the Pythagorean identity, factorization of sums of cubes, and substitution techniques. Ideal for Grades 11-12 or college-level students.

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