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To complete the proof of the identity \((1 + \tan x)^2 = \sec^2 x + 2 \tan x\), let’s break down the process step-by-step:

### Step 1: Expand \((1 + \tan x)^2\)
The first step involves expanding the square:
\[
(1 + \tan x)^2 = 1^2 + 2 \times 1 \times \tan x + (\tan x)^2 = 1 + 2 \tan x + \tan^2 x.
\]
This step uses the **binomial expansion** formula: \((a + b)^2 = a^2 + 2ab + b^2\). Therefore, the rule for this step is:
- **Rule 1**: **Binomial Expansion**.

### Step 2: Use a Trigonometric Identity
Next, we use the identity:
\[
1 + \tan^2 x = \sec^2 x.
\]
Substituting this into the equation gives:
\[
1 + 2 \tan x + \tan^2 x = \sec^2 x + 2 \tan x.
\]
This step involves applying the **Pythagorean Identity**: \(1 + \tan^2 x = \sec^2 x\). So, the rule for this step is:
- **Rule 2**: **Pythagorean Identity**.

### Summary of Steps
1. **Binomial Expansion**: \((1 + \tan x)^2 = 1 + 2 \tan x + \tan^2 x\).
2. **Pythagorean Identity**: \(1 + 2 \tan x + \tan^2 x = \sec^2 x + 2 \tan x\).

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the Pythagorean identity for sine and cosine functions?
2. How can the identity \(1 + \tan^2 x = \sec^2 x\) be derived?
3. What are some other common trigonometric identities used in proofs?
4. How would the expansion change if the expression were \((1 - \tan x)^2\)?
5. Can you think of a real-world application where this identity might be useful?

**Tip**: Knowing your basic trigonometric identities, like the Pythagorean identities, is crucial for simplifying and proving more complex trigonometric expressions.

Explore how to prove the trigonometric identity (1 + tan x)^2 = sec^2 x + 2 tan x using binomial expansion and the Pythagorean identity in trigonometry. This solution is suitable for students in grades 10-12 and covers key trigonometric and algebraic concepts.

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