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The given problem asks to find the value of the expression:

\[
7 \cdot \cot(2x) + 5
\]
where \( \tan(x) = \frac{8}{7} \).

### Step 1: Calculate \(\cot(x)\)
We know that:
\[
\cot(x) = \frac{1}{\tan(x)} = \frac{1}{\frac{8}{7}} = \frac{7}{8}.
\]

### Step 2: Use the double-angle formula for \(\cot(2x)\)
The double-angle identity for cotangent is:
\[
\cot(2x) = \frac{\cot^2(x) - 1}{2 \cdot \cot(x)}.
\]

Substitute \( \cot(x) = \frac{7}{8} \) into the formula:
\[
\cot(2x) = \frac{\left( \frac{7}{8} \right)^2 - 1}{2 \cdot \frac{7}{8}} = \frac{\frac{49}{64} - 1}{\frac{14}{8}} = \frac{\frac{49}{64} - \frac{64}{64}}{\frac{14}{8}} = \frac{-\frac{15}{64}}{\frac{14}{8}}.
\]

Simplify:
\[
\cot(2x) = \frac{-15}{64} \cdot \frac{8}{14} = \frac{-120}{896} = \frac{-15}{112}.
\]

### Step 3: Substitute \(\cot(2x)\) into the expression
Now, substitute \(\cot(2x) = \frac{-15}{112}\) into the original expression:
\[
7 \cdot \cot(2x) + 5 = 7 \cdot \frac{-15}{112} + 5 = \frac{-105}{112} + 5 = \frac{-105}{112} + \frac{560}{112} = \frac{455}{112}.
\]

### Step 4: Simplify the result
Finally, simplify the fraction:
\[
\frac{455}{112} = 4.0625.
\]

Thus, the value of the expression is \( 4.0625 \).

Would you like more details or clarifications on any step?

Here are 5 related questions you can explore:
1. What is the general identity for \(\cot(2x)\)?
2. How do you derive the double-angle identities for cotangent?
3. What is the relationship between \(\tan(x)\) and \(\cot(x)\)?
4. How would the result change if \(\tan(x) = \frac{7}{8}\)?
5. Can you solve similar expressions with different trigonometric identities?

**Tip**: Always double-check trigonometric identities when solving complex expressions to avoid mistakes.

Найдите значение выражения: 7ctg(2x) + 5, если tg(x) = 8/7

Learn how to solve the trigonometric expression 7ctg(2x) + 5 where tan(x) = 8/7. This solution uses double-angle formulas for cotangent and is suitable for high school students in Grades 10-12.

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