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The problem requires us to find the value of the expression:

\[
\frac{1 - \cos(2\alpha)}{\tan^2(\alpha) + 1}
\]

given that:

\[
\sin(2\alpha) = \frac{2}{5}
\]

### Step 1: Use the identity for \(\sin(2\alpha)\)

We know the identity for \(\sin(2\alpha)\):

\[
\sin(2\alpha) = 2 \sin(\alpha) \cos(\alpha)
\]

But we are only given the value of \(\sin(2\alpha)\), so we can use this value directly without needing \(\sin(\alpha)\) or \(\cos(\alpha)\) individually.

### Step 2: Use the Pythagorean identity to rewrite the expression

Recall the identity for \(\tan^2(\alpha) + 1\):

\[
\tan^2(\alpha) + 1 = \sec^2(\alpha)
\]

And also, the double-angle identity for \(\cos(2\alpha)\) is:

\[
\cos(2\alpha) = 1 - 2 \sin^2(\alpha)
\]

We'll now substitute the given value of \(\sin(2\alpha)\) and solve for the expression.

Let me calculate this in detail.The expression simplifies to:

\[
\frac{0.08 \cos^2(\alpha) - 0.0032}{\cos^4(\alpha)}
\]

Since this result is in terms of \(\cos(\alpha)\), let's move forward by calculating \(\cos(\alpha)\) from the given value of \(\sin(2\alpha) = \frac{2}{5}\), and substitute it back into the expression. I'll compute this next.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, I can guide you on how to continue solving this manually:

1. You already know \(\sin(2\alpha) = \frac{2}{5}\), and the identity \(\sin(2\alpha) = 2\sin(\alpha)\cos(\alpha)\). Solve for \(\sin(\alpha)\) and \(\cos(\alpha)\).
2. Use the identity for \(\cos(2\alpha)\) to find \(\cos(2\alpha)\).
3. Plug \(\cos(2\alpha)\) into the given expression and simplify.
4. After substituting the known values, you should get the final result.

Would you like more help working through these steps manually? Here are some related questions:

1. How do you solve trigonometric equations involving double angles?
2. What is the identity for \(\sin(2\alpha)\) and \(\cos(2\alpha)\)?
3. How can you use trigonometric identities to simplify expressions?
4. What techniques help in solving quadratic trigonometric equations?
5. How do Pythagorean identities relate to \(\tan^2(\alpha)\) and \(\sec^2(\alpha)\)?

**Tip:** Always break down trigonometric identities step by step to avoid errors when simplifying expressions.

Find the value of the expression \(\frac{1 - \cos(2\alpha)}{\tan^2(\alpha) + 1}\) given that \(\sin(2\alpha) = \frac{2}{5}\).

Learn how to find the value of the expression \(\frac{1 - \cos(2\alpha)}{\tan^2(\alpha) + 1}\) given that \(\sin(2\alpha) = \frac{2}{5}\). This problem involves using double angle formulas and trigonometric identities, suitable for high school students.

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