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Let's break down and solve the question visible in the image.

### Given:
1. \( 7 \tan(\alpha) = 1 \), so \( \tan(\alpha) = \frac{1}{7} \)
2. \( \sqrt{10 - 10 \cos(2\beta)} = \sqrt{2} \)

We are asked to find the value of \( \tan(\alpha) + \tan(2\beta) \), where \( \alpha, \beta \in \left( 0, \frac{\pi}{2} \right) \).

#### Step 1: Solve for \( \tan(2\beta) \)
We are given:
\[
\sqrt{10 - 10\cos(2\beta)} = \sqrt{2}
\]
Squaring both sides:
\[
10 - 10 \cos(2\beta) = 2
\]
\[
10 \cos(2\beta) = 8
\]
\[
\cos(2\beta) = \frac{4}{5}
\]
Now, using the identity:
\[
\tan(2\beta) = \frac{\sin(2\beta)}{\cos(2\beta)}
\]
We know:
\[
\sin^2(2\beta) + \cos^2(2\beta) = 1
\]
\[
\sin^2(2\beta) = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25}
\]
Thus:
\[
\sin(2\beta) = \frac{3}{5}
\]
Now:
\[
\tan(2\beta) = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}
\]

#### Step 2: Calculate \( \tan(\alpha) + \tan(2\beta) \)
From earlier, we know:
\[
\tan(\alpha) = \frac{1}{7}, \quad \tan(2\beta) = \frac{3}{4}
\]
Adding these:
\[
\tan(\alpha) + \tan(2\beta) = \frac{1}{7} + \frac{3}{4}
\]
Finding a common denominator:
\[
\tan(\alpha) + \tan(2\beta) = \frac{4 + 21}{28} = \frac{25}{28}
\]

Thus, the value of \( \tan(\alpha) + \tan(2\beta) \) is \( \frac{25}{28} \).

### Answer:
The correct option is (1) \( \frac{25}{28} \).

Would you like more details or have any questions?

Here are five related questions for practice:
1. What is the double angle identity for \( \tan(2\beta) \)?
2. How do you derive the value of \( \tan(\alpha) \) if \( 7\tan(\alpha) = 1 \)?
3. What is the value of \( \cos(2\beta) \) when \( \sqrt{10 - 10 \cos(2\beta)} = \sqrt{2} \)?
4. How do you calculate \( \tan(\alpha) + \tan(2\beta) \) if \( \alpha \) and \( \beta \) are given in terms of radians?
5. What is the sum of angles formula for tangent?

**Tip**: Always use trigonometric identities to simplify expressions before plugging in values.

If 7tan(α) = 1 and √(10 - 10cos(2β)) = √2, with α, β ∈ (0, π/2), then the value of tan(α) + tan(2β) is:

Learn how to solve for tan(α) + tan(2β) using trigonometric identities and algebraic methods. The problem involves using the double angle identity for tangent, solving quadratic equations, and applying Pythagorean identities. Suitable for students in Grades 11-12.

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