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To solve this problem, we need to express the given trigonometric expression \(\tan^2(x) + 2 \csc(x) \tan(x)\) in terms of \(a = \sec(x)\) and \(b = \csc(x)\).

### Step 1: Simplify the Expression Using Trigonometric Identities
We know:
\[
\sec(x) = \frac{1}{\cos(x)} \quad \text{and} \quad \csc(x) = \frac{1}{\sin(x)}
\]
Given:
\[
\sec(x) = a \quad \text{and} \quad \csc(x) = b
\]

### Step 2: Express \(\tan(x)\) in Terms of \(a\) and \(b\)
\[
\tan(x) = \frac{\sin(x)}{\cos(x)} = \frac{1}{\csc(x)\sec(x)} = \frac{1}{ab}
\]
So,
\[
\tan(x) = \frac{1}{ab}
\]

### Step 3: Substitute into the Given Expression
Now substitute \(\tan(x)\) and \(\csc(x)\) into the expression \(\tan^2(x) + 2 \csc(x) \tan(x)\):
\[
\tan^2(x) + 2 \csc(x) \tan(x) = \left(\frac{1}{ab}\right)^2 + 2b\left(\frac{1}{ab}\right)
\]

### Step 4: Simplify the Expression
\[
\left(\frac{1}{ab}\right)^2 = \frac{1}{a^2b^2}
\]
\[
2b\left(\frac{1}{ab}\right) = \frac{2}{a}
\]
Thus:
\[
\frac{1}{a^2b^2} + \frac{2}{a}
\]
Finally, find a common denominator:
\[
\frac{1}{a^2b^2} + \frac{2}{a} = \frac{1 + 2a^2b^2}{a^2b^2}
\]

So, the expression \(\tan^2(x) + 2 \csc(x) \tan(x)\) in terms of \(a\) and \(b\) is:
\[
\boxed{\frac{1 + 2a^2b^2}{a^2b^2}}
\]

Would you like more details on any of these steps or have any questions?

Here are some related questions:
1. How can we express \(\sin(x)\) and \(\cos(x)\) in terms of \(a\) and \(b\)?
2. What is the significance of using trigonometric identities in simplifying expressions?
3. How would you approach expressing \(\cot(x)\) in terms of \(a\) and \(b\)?
4. Can we generalize this approach to other trigonometric identities?
5. How can these simplifications help in solving trigonometric equations?

**Tip:** Always remember to double-check the relationships between trigonometric functions when substituting values to ensure accuracy in your expressions.

If sec(x) = a and csc(x) = b, use trigonometric identities to express tan^2(x) + 2csc(x)tan(x) in terms of a and b.

Learn how to simplify and express tan^2(x) + 2csc(x)tan(x) in terms of sec(x) = a and csc(x) = b using trigonometric identities. This detailed solution is suitable for high school students tackling advanced trigonometric problems.

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