The problem involves verifying a trigonometric identity by identifying the rules that justify each step.

Step 1: Expansion of $(1 + \cot x)^2$

• The statement $(1 + \cot x)^2 = 1 + 2 \cot x + \cot^2 x$ uses the Algebra Rule (specifically, the expansion of a binomial).

Step 2: Simplification to $\csc^2 x + 2 \cot x$

• The step $1 + \cot^2 x = \csc^2 x$ relies on the Pythagorean Identity, which states $1 + \cot^2 x = \csc^2 x$.

Rules for Each Step:

1. Algebra: Expanding the binomial $(1 + \cot x)^2$.
2. Pythagorean: Simplifying $1 + \cot^2 x$ to $\csc^2 x$.

Would you like further clarification or additional examples to reinforce these concepts?

Here are 5 questions to deepen your understanding:

1. What is the Pythagorean identity involving cotangent and cosecant?
2. How do you expand any binomial squared expression?
3. Can you explain why $1 + \cot^2 x = \csc^2 x$ holds true geometrically?
4. What are some other identities derived from the Pythagorean theorem in trigonometry?
5. How would the process change if verifying a different identity like $\sec^2 x - 1 = \tan^2 x$?

Tip: Always start by identifying which trigonometric identities and algebraic rules apply in a proof to simplify each step logically.