verify tan^2(x)(1+cot^2(x)) = 1/(1-sin^2(x)) ... list the names the properties using in each step, examples of acceptable property names are things such as reciprocal identity, distributive property, quotient identity, etc.

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.

---

### 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**: \( \cos^2(x) = 1 - \sin^2(x) \)

---

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!

Verify tan^2(x)(1 + cot^2(x)) = 1/(1 - sin^2(x)). List the names of the properties used in each step.

Step-by-step solution to verify the trigonometric identity tan^2(x)(1 + cot^2(x)) = 1/(1 - sin^2(x)). Learn how to use key trigonometric identities such as the Pythagorean identity and reciprocal identities in this trigonometry problem. Suitable for high school students in Grades 10-12.

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