Let's verify the identity $\tan^2(x)(1 + \cot^2(x)) = \frac{1}{1 - \sin^2(x)}$ step-by-step, naming the properties used in each step.

Step 1: Start with the left-hand side (LHS)

$\tan^2(x)(1 + \cot^2(x))$

Property Used: Definition of cotangent

We use the identity $\cot(x) = \frac{1}{\tan(x)}$, so: $1 + \cot^2(x) = 1 + \frac{1}{\tan^2(x)}$ Thus, the expression becomes: $\tan^2(x)\left(1 + \frac{1}{\tan^2(x)}\right)$

Step 2: Simplify the expression inside the parentheses

Property Used: Common denominator

To combine terms inside the parentheses, we find a common denominator: $1 + \frac{1}{\tan^2(x)} = \frac{\tan^2(x) + 1}{\tan^2(x)}$

Step 3: Multiply $\tan^2(x)$ by the result of the parentheses

$\tan^2(x) \cdot \frac{\tan^2(x) + 1}{\tan^2(x)}$

Property Used: Cancellation property

Here, $\tan^2(x)$ cancels with the $\tan^2(x)$ in the denominator: $\tan^2(x) + 1$

Step 4: Use the Pythagorean identity

The identity $1 + \tan^2(x) = \sec^2(x)$ allows us to rewrite the expression: $\sec^2(x)$

Step 5: Express $\sec^2(x)$ in terms of $\sin(x)$

Property Used: Reciprocal identity

Recall that $\sec(x) = \frac{1}{\cos(x)}$, so: $\sec^2(x) = \frac{1}{\cos^2(x)}$

Step 6: Use the Pythagorean identity for sine and cosine

We know that $\cos^2(x) = 1 - \sin^2(x)$, so: $\sec^2(x) = \frac{1}{1 - \sin^2(x)}$

This is the right-hand side (RHS) of the equation, so the identity is verified.

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Conclusion:

The identity is verified as: $\tan^2(x)(1 + \cot^2(x)) = \frac{1}{1 - \sin^2(x)}$

Properties used:

1. Definition of cotangent: $\cot(x) = \frac{1}{\tan(x)}$
2. Common denominator: Combining terms with the same denominator
3. Cancellation property: Canceling common factors in numerator and denominator
4. Pythagorean identity: $1 + \tan^2(x) = \sec^2(x)$
5. Reciprocal identity: $\sec(x) = \frac{1}{\cos(x)}$
6. Pythagorean identity for sine and cosine: $\cos^2(x) = 1 - \sin^2(x)$

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Would you like further details or explanations on any step?

Here are five related questions for further practice:

1. Verify $1 + \cot^2(x) = \csc^2(x)$.
2. Show that $\sin^2(x) + \cos^2(x) = 1$.
3. Simplify $\frac{1}{1 - \sin^2(x)}$.
4. Prove $\tan(x) = \frac{\sin(x)}{\cos(x)}$ using the quotient identity.
5. Verify $\frac{1}{\sec^2(x)} = \cos^2(x)$.

Tip: Always try to recognize common trigonometric identities when simplifying expressions!