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Điều kiện alpha khác kπ,k∈Z để làm gì

The image contains some standard trigonometric identities, and the condition \( \alpha \neq k\pi \) (where \( k \in \mathbb{Z} \), the set of integers) is a restriction on the variable \( \alpha \) to avoid certain undefined points or singularities. Here's why this condition is important for each equation:

1. **First equation:**
   \[
   \tan \alpha \cdot \cot \alpha = 1 \quad \text{với} \quad \alpha \neq k \frac{\pi}{2}, \, k \in \mathbb{Z}
   \]
   - The restriction \( \alpha \neq k \frac{\pi}{2} \) ensures that \( \tan \alpha \) and \( \cot \alpha \) are both defined. At \( \alpha = k \frac{\pi}{2} \), the tangent and cotangent functions have vertical asymptotes (they are undefined).

2. **Second equation:**
   \[
   1 + \tan^2 \alpha = \frac{1}{\cos^2 \alpha} \quad \text{với} \quad \alpha \neq k\pi, \, k \in \mathbb{Z}
   \]
   - This is the Pythagorean identity. The restriction \( \alpha \neq k\pi \) ensures \( \cos \alpha \neq 0 \), since the cosine function equals zero at \( \alpha = \frac{\pi}{2} + k\pi \). This condition prevents division by zero in the equation.

3. **Third equation:**
   \[
   \alpha \neq \frac{\pi}{2} + k\pi, \, k \in \mathbb{Z} \quad 1 + \cot^2 \alpha = \frac{1}{\sin^2 \alpha}
   \]
   - This is another Pythagorean identity, but for cotangent. The condition \( \alpha \neq \frac{\pi}{2} + k\pi \) ensures that \( \sin \alpha \neq 0 \), preventing division by zero (since \( \sin \alpha = 0 \) at these values).

### Purpose of the Condition \( \alpha \neq k\pi \):
- The restriction ensures that the trigonometric functions (tangent, cotangent, cosine, and sine) are defined and avoid undefined behavior like division by zero or encountering points where the functions do not exist (singularities).

Let me know if you'd like further clarification or details.

### Related Questions:
1. What happens to the tangent and cotangent functions at \( \alpha = k\frac{\pi}{2} \)?
2. How are these trigonometric identities derived from the unit circle?
3. What is the significance of singularities in trigonometric functions?
4. Can the Pythagorean identities be used to solve for angles in specific quadrants?
5. How can you generalize the values of \( \alpha \) that cause these trigonometric functions to be undefined?

### Tip:
Always check the domain of trigonometric functions before applying identities to avoid errors in computation due to undefined values.

Explore key trigonometric identities including tangent, cotangent, sine, and cosine functions. Learn why domain restrictions like α ≠ kπ and α ≠ π/2 + kπ are important in trigonometry to prevent undefined behavior.

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