4sin^2 (x) + 4cos^2 (x) - 9tan(x) * cot(x)

To simplify the expression:

\[
4\sin^2(x) + 4\cos^2(x) - 9\tan(x) \cdot \cot(x)
\]

we can apply trigonometric identities step by step:

### Step 1: Simplify \(\sin^2(x) + \cos^2(x)\)

We know that:

\[
\sin^2(x) + \cos^2(x) = 1
\]

Therefore:

\[
4\sin^2(x) + 4\cos^2(x) = 4(\sin^2(x) + \cos^2(x)) = 4 \times 1 = 4
\]

### Step 2: Simplify \(\tan(x) \cdot \cot(x)\)

We also know that:

\[
\tan(x) = \frac{\sin(x)}{\cos(x)}, \quad \cot(x) = \frac{\cos(x)}{\sin(x)}
\]

Thus:

\[
\tan(x) \cdot \cot(x) = \left(\frac{\sin(x)}{\cos(x)}\right) \cdot \left(\frac{\cos(x)}{\sin(x)}\right) = \frac{\sin(x) \cos(x)}{\cos(x) \sin(x)} = 1
\]

So:

\[
-9 \tan(x) \cdot \cot(x) = -9 \times 1 = -9
\]

### Step 3: Combine the results

Combining these results gives:

\[
4 - 9 = -5
\]

### Final Simplified Expression:

\[
\boxed{-5}
\]

Would you like more details, or do you have any other questions?

### Related Questions:

1. What are some common trigonometric identities?
2. How can you simplify more complex trigonometric expressions?
3. What is the significance of the Pythagorean identity in trigonometry?
4. How do tangent and cotangent relate to sine and cosine?
5. What are some applications of trigonometric identities in calculus?

### Tip: 
Always check for fundamental trigonometric identities like \(\sin^2(x) + \cos^2(x) = 1\) when simplifying expressions; they can simplify the problem significantly!

Discover how to simplify the trigonometric expression 4sin^2(x) + 4cos^2(x) - 9tan(x) * cot(x) using fundamental trigonometric identities like the Pythagorean identity. This step-by-step guide is suitable for students in Grades 10-12.

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