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This page contains trigonometric identities and proofs. Here's a breakdown of the problems on the page:

1. **Problem 22**: Prove the identities:
   - \( \frac{\tan \alpha + \cot \alpha}{\sin \alpha \cdot \cos \alpha} = 1 \)
   - \( \frac{\tan \alpha - 1}{\tan \alpha + 1} = \frac{\sin^2 \alpha - \cos^2 \alpha}{\sin \alpha - \cos \alpha} \)
   - \( \frac{\tan \alpha}{1 + \tan^2 \alpha} = \sin \alpha \cdot \cos \alpha \)

2. **Problem 23**: Prove the identities:
   - \( (\sin x + \cos x) \cdot (\sin x + \cos x) = (1 + \sin x) \cdot (1 + \cos x) \)
   - \( \sin x + \cos x \cdot \sin x + \cos x = \frac{1}{\sin x} + \frac{1}{\cos x} \)

3. **Problem 24**: Prove the identities:
   - \( (1 - \cos \theta)^2 + (1 - \sin \theta)^2 = \left( \frac{1}{\cos \theta \cdot \sin \theta} \right)^2 \)
   - \( 2\theta + \sin^2 \theta = (\cos \theta + 1)^2 \)

4. **Problem 25**: Prove the identities:
   - \( \frac{1}{\sqrt{\sin^2 x}} + \frac{1}{\sqrt{\cos^2 x}} = \sin x + \cos x \) for \( x \in \left( 0, \frac{\pi}{2} \right) \)
   - \( \frac{1 - \sin x}{1 + \sin x} = \frac{\cos x}{1 + \cos x} \) for \( x \in \left( 0, \frac{\pi}{2} \right) \)

5. **Problem 26**: Prove the identities:
   - \( \sin^4 x + \cos^4 x = 1 - 2\sin^2 x \cdot \cos^2 x \) for all \( x \in \mathbb{R} \)
   - \( \sin^6 x + \cos^6 x = 1 - 3\sin^2 x \cdot \cos^2 x \) for all \( x \in \mathbb{R} \)
   - Prove that the expression \( \sin^4 x + \cos^4 x - 2 \sin^4 x \cdot \cos^4 x + \sin^6 x - \cos^4 x + \sin^4 x \cdot \cos^6 x \) is constant, indicating it does not depend on \( x \).

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### Additional Related Questions
1. How can we approach proving trigonometric identities systematically?
2. What are common simplifications used in trigonometric identities involving products of sine and cosine?
3. Why does using Pythagorean identities often help in simplifying trigonometric expressions?
4. How do we handle identities involving sums and differences of sine and cosine?
5. What role do reciprocal identities play in simplifying expressions involving tangent and cotangent?

### Tip:
To simplify trigonometric identities, try expressing all terms in terms of sine and cosine first.

Trigonometric identity proofs involving sine, cosine, and tangent functions, as shown in the uploaded image.

Explore trigonometric identity proofs involving sine, cosine, and tangent functions. This page covers identities essential for high school algebra, including Pythagorean identities, reciprocal identities, and higher powers of sine and cosine.

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