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do by splitting like this tan^2 (2x) * tan(2x) * sec(2x)and use sec^2(2x)-tan^2(2x)=1 formula

The given expression is:

\[
\text{tan}^3(2x) \cdot \sec(2x)
\]

You suggested splitting it as:

\[
\left( \tan^2(2x) \right) \cdot \tan(2x) \cdot \sec(2x)
\]

Now, using the identity:

\[
\sec^2(2x) - \tan^2(2x) = 1
\]

We can rearrange this identity to express \(\tan^2(2x)\) in terms of \(\sec^2(2x)\):

\[
\tan^2(2x) = \sec^2(2x) - 1
\]

Substitute this into the expression:

\[
(\sec^2(2x) - 1) \cdot \tan(2x) \cdot \sec(2x)
\]

Now expand this:

\[
\left( \sec^2(2x) - 1 \right) \cdot \tan(2x) \cdot \sec(2x)
= \sec^2(2x) \cdot \tan(2x) \cdot \sec(2x) - \tan(2x) \cdot \sec(2x)
\]

Simplifying each term:

1. \(\sec^2(2x) \cdot \sec(2x) \cdot \tan(2x) = \sec^3(2x) \cdot \tan(2x)\)
2. \(\tan(2x) \cdot \sec(2x)\) remains as it is.

So the expression becomes:

\[
\sec^3(2x) \cdot \tan(2x) - \sec(2x) \cdot \tan(2x)
\]

You can factor out \(\sec(2x) \cdot \tan(2x)\):

\[
\sec(2x) \cdot \tan(2x) \left( \sec^2(2x) - 1 \right)
\]

And since \(\sec^2(2x) - 1 = \tan^2(2x)\), we have:

\[
\sec(2x) \cdot \tan(2x) \cdot \tan^2(2x)
= \sec(2x) \cdot \tan^3(2x)
\]

Thus, the simplified form of the expression is:

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

Would you like further details or have any questions?

Here are some related questions for you:

1. What is the derivative of \(\sec(2x)\) and \(\tan(2x)\)?
2. How do you integrate \(\sec^3(x)\)?
3. What is the physical meaning of the secant and tangent functions?
4. Can the identity \(\sec^2(2x) - \tan^2(2x) = 1\) be applied to any angle?
5. How can the Pythagorean identity be generalized for other trigonometric functions?

**Tip**: When dealing with multiple trigonometric identities, it's helpful to first simplify the expression as much as possible using basic identities.

do by splitting like this tan^2(2x) * tan(2x) * sec(2x) and use sec^2(2x) - tan^2(2x) = 1 formula

Step-by-step solution for simplifying tan^3(2x) sec(2x) using the identity sec^2(2x) - tan^2(2x) = 1. This problem involves key trigonometric concepts and is suitable for high school students studying trigonometry in Grades 10-12.

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