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We are given the expression \( 6 \cos(2x) - 4 \) and the condition \( \tan(x) = -\frac{10}{3} \). We need to find the value of the expression based on this information.

### Step 1: Understanding the tangent relationship
From the identity for tangent, we know that:
\[
\tan(x) = \frac{\sin(x)}{\cos(x)}
\]
So, if \( \tan(x) = -\frac{10}{3} \), we can construct a right triangle where the opposite side is \(-10\) and the adjacent side is \(3\). Using the Pythagorean theorem, we can find the hypotenuse:
\[
\text{Hypotenuse} = \sqrt{(-10)^2 + 3^2} = \sqrt{100 + 9} = \sqrt{109}
\]

Thus, for this angle \(x\), we have:
\[
\sin(x) = \frac{-10}{\sqrt{109}}, \quad \cos(x) = \frac{3}{\sqrt{109}}
\]

### Step 2: Using the double-angle identity for cosine
We need to find \( \cos(2x) \). The double-angle identity for cosine is:
\[
\cos(2x) = \cos^2(x) - \sin^2(x)
\]
Substituting the values for \( \cos(x) \) and \( \sin(x) \):
\[
\cos(2x) = \left( \frac{3}{\sqrt{109}} \right)^2 - \left( \frac{-10}{\sqrt{109}} \right)^2
\]
\[
\cos(2x) = \frac{9}{109} - \frac{100}{109} = \frac{-91}{109}
\]

### Step 3: Substituting into the original expression
Now, we substitute \( \cos(2x) = \frac{-91}{109} \) into the expression \( 6 \cos(2x) - 4 \):
\[
6 \cos(2x) - 4 = 6 \left( \frac{-91}{109} \right) - 4 = \frac{-546}{109} - 4
\]
Convert 4 to have the same denominator:
\[
4 = \frac{436}{109}
\]
Thus:
\[
6 \cos(2x) - 4 = \frac{-546 - 436}{109} = \frac{-982}{109}
\]

### Final Answer:
The value of the expression is \( \frac{-982}{109} \), which simplifies to approximately \( -9.01 \).

Would you like further details on any steps? Here are some related questions you might find useful:

1. How do you derive the double-angle formula for cosine?
2. What is the relationship between tangent and sine/cosine in right triangles?
3. How do you find trigonometric function values from triangle dimensions?
4. What are the other double-angle identities in trigonometry?
5. How would this process change if the tangent were positive?

**Tip:** When dealing with trigonometric identities, always keep in mind the quadrant of the angle to correctly assign signs to the sine and cosine values.

Найдите значение выражения: 6 cos(2x) - 4, если tg(x) = -10/3

Solve a trigonometric expression involving a double-angle identity. Given tan(x) = -10/3, we calculate the value of 6cos(2x) - 4 using trigonometric identities and the Pythagorean theorem. Suitable for Grades 10-12.

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