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?

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.