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We are given the expression:

\[
\frac{\tan \beta + \cot \beta}{\tan \beta} = \frac{\tan \beta}{\tan \beta} + \frac{\cot \beta}{\tan \beta}
\]

Our goal is to simplify this expression using trigonometric identities and show that it equals \(\csc^2 \beta\).

### Step-by-Step Solution:

1. **Simplify the first term:**
   \[
   \frac{\tan \beta}{\tan \beta} = 1
   \]
   This simplifies the equation to:
   \[
   1 + \frac{\cot \beta}{\tan \beta}
   \]

2. **Simplify the second term:**
   Recall the identity for cotangent:
   \[
   \cot \beta = \frac{1}{\tan \beta}
   \]
   Using this identity, we substitute:
   \[
   \frac{\cot \beta}{\tan \beta} = \frac{\frac{1}{\tan \beta}}{\tan \beta} = \frac{1}{\tan^2 \beta}
   \]

   Now the expression becomes:
   \[
   1 + \frac{1}{\tan^2 \beta}
   \]

3. **Use the Pythagorean identity:**
   The identity \(\csc^2 \beta = 1 + \cot^2 \beta\) can be rewritten using \(\cot \beta = \frac{1}{\tan \beta}\):
   \[
   1 + \frac{1}{\tan^2 \beta} = \csc^2 \beta
   \]

Thus, the final expression simplifies to:
\[
\csc^2 \beta
\]

This shows that the left side of the equation is equal to the right side, as required.

---

Would you like a more detailed explanation on any part of the process, or any specific questions? Here are 5 related questions:

1. How is the cotangent function defined in terms of tangent and sine?
2. Can you explain why \(\cot \beta = \frac{1

Use trigonometric identities to transform the left side of the equation into the right side (0 < θ < π/2).

This problem focuses on simplifying a trigonometric identity using known formulas such as cotangent and tangent definitions, aiming to show that tan(β) + cot(β) / tan(β) equals csc^2(β). It demonstrates core trigonometry concepts relevant for high school students (Grades 9-12).

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