https://file.aiquickdraw.com/mathbot/file/1726708536414itiwubpr.jpg

To prove the identity:

\[
(\sin x + \cos x)^2 = 1 + 2 \sin x \cos x
\]

We can follow these steps:

### Step 1: Expand the left-hand side
Using the algebraic identity \((a + b)^2 = a^2 + 2ab + b^2\), expand the expression:

\[
(\sin x + \cos x)^2 = \sin^2 x + 2 \sin x \cos x + \cos^2 x
\]

### Step 2: Apply the Pythagorean Identity
Recall the Pythagorean identity:

\[
\sin^2 x + \cos^2 x = 1
\]

Using this, substitute into the expanded form:

\[
1 + 2 \sin x \cos x
\]

### Step 3: Conclusion
This gives the right-hand side of the original equation. Therefore, the identity is proven:

\[
(\sin x + \cos x)^2 = 1 + 2 \sin x \cos x
\]

If this is what you were trying to input in the tool shown in the image, the steps are as follows:

1. Start with \((\sin x + \cos x)^2\).
2. Apply the rule of expansion: \( \sin^2 x + 2 \sin x \cos x + \cos^2 x\).
3. Apply the Pythagorean identity to simplify \( \sin^2 x + \cos^2 x\) into 1.
4. The final expression becomes \( 1 + 2 \sin x \cos x\).

Would you like further clarification on any part of this process? 

Here are 5 follow-up questions to deepen your understanding:
1. What other trigonometric identities are similar to the Pythagorean identity?
2. How would the process change if we were proving a different identity involving tangent or secant?
3. What happens if we expand \((\sin x - \cos x)^2\)?
4. How can this identity be applied in solving trigonometric equations?
5. Can we use this identity to derive any other important trigonometric formulas?

**Tip:** Always remember to use fundamental identities like the Pythagorean identity when simplifying trigonometric expressions.

Prove the identity: (sin(x) + cos(x))^2 = 1 + 2 sin(x) cos(x)

Learn how to prove the trigonometric identity (sin(x) + cos(x))^2 = 1 + 2 sin(x) cos(x) using algebraic expansion and the Pythagorean identity. This detailed solution is perfect for students in Grades 10-12 studying trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation