To simplify the expression \((3Tg10º + 8Cot80º)Cot10º\), let's start by simplifying each trigonometric term separately.

First, we know the following identities:
1. \(\tan(90º - x) = \cot(x)\)
2. \(\cot(90º - x) = \tan(x)\)

Given \( \cot(80º) = \tan(10º) \), we can rewrite the expression using these identities:
\[ 8 \cot(80º) = 8 \tan(10º) \]

Thus, the expression becomes:
\[ (3 \tan(10º) + 8 \tan(10º)) \cot(10º) \]

Combining the like terms inside the parenthesis:
\[ (11 \tan(10º)) \cot(10º) \]

We know that:
\[ \cot(x) = \frac{1}{\tan(x)} \]

Therefore:
\[ (11 \tan(10º)) \cdot \cot(10º) = 11 \tan(10º) \cdot \frac{1}{\tan(10º)} = 11 \]

Thus, the simplified form of the expression \((3 \tan(10º) + 8 \cot(80º)) \cot(10º)\) is:
\[ 11 \]

Would you like more details or have any questions?

Here are 8 related questions for further exploration:
1. What is the value of \(\tan(10º)\)?
2. How do you derive the identity \(\tan(90º - x) = \cot(x)\)?
3. What are the properties of the cotangent function?
4. Can you provide a proof for the identity \(\cot(x) = \frac{1}{\tan(x)}\)?
5. How would the expression change if it involved \(\tan(45º)\) instead?
6. What is the value of \(\cot(10º)\)?
7. How do you convert between degrees and radians in trigonometry?
8. What are the applications of these trigonometric identities in real-world problems?

**Tip:** When working with trigonometric identities, always look for opportunities to simplify the expression by converting one trigonometric function into another using known identities.

Learn how to simplify the trigonometric expression (3Tg10º + 8Cot80º)Cot10º step-by-step. This problem involves applying trigonometric identities and properties, including the cotangent and tangent functions. Suitable for high school students studying trigonometry.

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