Math Problem Statement

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

Solution

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

Given equation:

sin(x)+sin2(x)+cos2(x)=1\sin(x) + \sin^2(x) + \cos^2(x) = 1

Step 1: Use the Pythagorean identity

One of the fundamental trigonometric identities is: sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1

Step 2: Substitute the identity into the equation

Replace sin2(x)+cos2(x)\sin^2(x) + \cos^2(x) with 1: sin(x)+(sin2(x)+cos2(x))=1\sin(x) + (\sin^2(x) + \cos^2(x)) = 1 sin(x)+1=1\sin(x) + 1 = 1

Step 3: Simplify the equation

Now, subtract 1 from both sides: sin(x)=0\sin(x) = 0

Final Answer:

The equation sin(x)+sin2(x)+cos2(x)=1\sin(x) + \sin^2(x) + \cos^2(x) = 1 simplifies to sin(x)=0\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\sin(x) = 0 in terms of xx?
  2. How is the Pythagorean identity derived?
  3. Can sin(x)+cos2(x)\sin(x) + \cos^2(x) equal any value other than 1?
  4. How do you solve sin2(x)+cos2(x)=1\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!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Pythagorean Identity

Formulas

sin^2(x) + cos^2(x) = 1

Theorems

Pythagorean Theorem in Trigonometry

Suitable Grade Level

Grades 10-12