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Let's solve this step by step using trigonometric identities.

We are tasked with transforming the left side of the equation into \( \csc^2 \beta \) using identities.

### Given Equation:
\[
\frac{\tan \beta + \cot \beta}{\tan \beta}
\]

### Step 1: Break the expression into parts.
Rewrite the left-hand side as two terms:
\[
\frac{\tan \beta}{\tan \beta} + \frac{\cot \beta}{\tan \beta}
\]

The first term simplifies immediately:
\[
1 + \frac{\cot \beta}{\tan \beta}
\]

### Step 2: Use trigonometric identity for \( \cot \beta \) and \( \tan \beta \).
Recall the relationship:
\[
\cot \beta = \frac{1}{\tan \beta}
\]
Thus, we can substitute into the equation:
\[
1 + \frac{\frac{1}{\tan \beta}}{\tan \beta} = 1 + \frac{1}{\tan^2 \beta}
\]

### Step 3: Apply the identity for \( \csc^2 \beta \).
From the Pythagorean identity:
\[
1 + \frac{1}{\tan^2 \beta} = \csc^2 \beta
\]

### Final Expression:
The original equation simplifies to:
\[
\csc^2 \beta
\]

Thus, we have successfully transformed the left-hand side of the equation into \( \csc^2 \beta \), matching the desired right-hand side.

Would you like more details on any step or have any questions?

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Here are 5 related questions to expand on this topic:
1. What are other common trigonometric identities used in equation transformations?
2. How can the Pythagorean identity be derived from basic trigonometric functions?
3. Can you derive \( \csc^2 \beta \) in terms of cosine and sine?
4. How does the tangent-cotangent identity relate to secant and cosecant?
5. What is the geometric interpretation of the cosecant function?

**Tip:** Memorizing the core trigonometric identities, especially Pythagorean identities, can greatly help in solving complex expressions quickly.

Use trigonometric identities to transform the left side of the equation into the right side (0 < θ < π/2). tan(β) + cot(β) / tan(β) = csc²(β)

Learn how to use trigonometric identities to transform expressions like tan(β) + cot(β) / tan(β) into csc²(β). This step-by-step solution covers key trigonometric concepts and identities, suitable for high school students in Grades 10-12.

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