Using a Pythagorean identity we see that the equation sin(x) + sin2(x) + cos2(x) = 1 is equivalent to the basic equation

To simplify the given equation using the Pythagorean identity, let's break it down step by step.

### Given equation:
\[
\sin(x) + \sin^2(x) + \cos^2(x) = 1
\]

### Step 1: Use the Pythagorean identity
One of the fundamental trigonometric identities is:
\[
\sin^2(x) + \cos^2(x) = 1
\]

### Step 2: Substitute the identity into the equation
Replace \(\sin^2(x) + \cos^2(x)\) with 1:
\[
\sin(x) + (\sin^2(x) + \cos^2(x)) = 1
\]
\[
\sin(x) + 1 = 1
\]

### Step 3: Simplify the equation
Now, subtract 1 from both sides:
\[
\sin(x) = 0
\]

### Final Answer:
The equation \(\sin(x) + \sin^2(x) + \cos^2(x) = 1\) simplifies to \(\sin(x) = 0\).

Would you like further details or have any questions about this?

Here are 5 related questions:
1. What are the solutions to \(\sin(x) = 0\) in terms of \(x\)?
2. How is the Pythagorean identity derived?
3. Can \(\sin(x) + \cos^2(x)\) equal any value other than 1?
4. How do you solve \(\sin^2(x) + \cos^2(x) = 1\) for complex values?
5. What are other common trigonometric identities that can simplify equations?

**Tip:** Always remember to look for fundamental identities like the Pythagorean identity when simplifying trigonometric equations!

Using a Pythagorean identity we see that the equation sin(x) + sin^2(x) + cos^2(x) = 1 is equivalent to the basic equation

This article provides a step-by-step solution to simplify the trigonometric equation sin(x) + sin^2(x) + cos^2(x) = 1 using the Pythagorean identity. Learn how the identity simplifies the equation to sin(x) = 0 and explore related concepts in trigonometry. Suitable for students in Grades 10-12.

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