It seems like the layout and phrasing of the question might have been jumbled. Allow me to clarify and address the math problem following the given information and logical steps.
Given:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
Proof:
[tex]\[ \cos^2(\theta) + \sin^2(\theta) = 1 \][/tex]
We want to show that the given relationships hold consistently through the statements.
1. Given:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
This is our initial equation, given in the problem. Here, [tex]\( r \)[/tex] can be thought of as the radius in a coordinate system.
2. Rewrite in the form of trigonometric identities:
[tex]\[ \frac{x^2}{r^2} + \frac{y^2}{r^2} = \frac{r^2}{r^2} \][/tex]
Dividing both sides of the given equation by [tex]\( r^2 \)[/tex], simplifies to:
[tex]\[ \frac{x^2}{r^2} + \frac{y^2}{r^2} = 1 \][/tex]
3. Transform into trigonometric functions:
[tex]\[ \left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2 = 1 \][/tex]
This step uses the trigonometric expressions where [tex]\( \cos(\theta) = \frac{x}{r} \)[/tex] and [tex]\( \sin(\theta) = \frac{y}{r} \)[/tex].
4. Define trigonometric functions in terms of [tex]\( x, y, \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[ \cos(\theta) = \frac{x}{r} \][/tex]
[tex]\[ \sin(\theta) = \frac{y}{r} \][/tex]
5. Square the trigonometric identities and sum them:
[tex]\[ \cos^2(\theta) + \sin^2(\theta) = \left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2 \][/tex]
By previously derived identities, this simplifies to:
[tex]\[ \cos^2(\theta) + \sin^2(\theta) = 1 \][/tex]
Conclusion:
The given statements follow logically and demonstrate that [tex]\( x^2 + y^2 = r^2 \)[/tex] translates through trigonometric identities to show that [tex]\( \cos^2(\theta) + \sin^2(\theta) = 1 \)[/tex].
Thus, the correct proof confirms the fundamental trigonometric identity.
