Select the correct answer from each drop-down menu.

Given: [tex]x^2 + y^2 = r^2[/tex]

Prove: [tex]\cos^2(\theta) + \sin^2(\theta) = 1[/tex]

\begin{tabular}{|l|l|}
\hline
Statements & Reasons \\
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1. [tex]x^2 + y^2 = r^2[/tex] & 1. Given \\
\hline
2. [tex]\frac{x^2}{r^2} + \frac{y^2}{r^2} = \frac{r^2}{r^2}[/tex] & 2. Divide both sides by [tex]r^2[/tex] \\
\hline
3. [tex]\left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2 = 1[/tex] & 3. Simplify \\
\hline
4. [tex]\cos^2(\theta) + \sin^2(\theta) = 1[/tex] & 4. Definitions of cosine and sine \\
\hline
\end{tabular}



Answer :

To prove [tex]\(\cos^2(\theta) + \sin^2(\theta) = 1\)[/tex] from the given equation [tex]\(x^2 + y^2 = r^2\)[/tex], follow these steps:

1. Statements: [tex]\(x^2 + y^2 = r^2\)[/tex]
Reasons: Given

2. Statements: [tex]\(\frac{x^2}{r^2} + \frac{y^2}{r^2} = \frac{r^2}{r^2}\)[/tex]
Reasons: Divide each term by [tex]\(r^2\)[/tex]

3. Statements: [tex]\(\left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2 = 1\)[/tex]
Reasons: Simplified

Continuing from the simplified equation:

- We know from trigonometry that:
[tex]\(x = r \cos(\theta)\)[/tex] and [tex]\(y = r \sin(\theta)\)[/tex]

- Substituting these values into the equation:
[tex]\[ \left(\frac{r \cos(\theta)}{r}\right)^2 + \left(\frac{r \sin(\theta)}{r}\right)^2 = 1 \][/tex]

- This further simplifies to:
[tex]\[ \cos^2(\theta) + \sin^2(\theta) = 1 \][/tex]

Thus, we have proven that [tex]\(\cos^2(\theta) + \sin^2(\theta) = 1\)[/tex].

[tex]\[ \begin{array}{|l|l|} \hline \text{Statements} & \text{Reasons} \\ \hline 1. x^2 + y^2 = r^2 & \text{1. Given} \\ \hline 2. \frac{x^2}{r^2} + \frac{y^2}{r^2} = \frac{r^2}{r^2} & \text{2. Divide each term by } r^2 \\ \hline 3. \left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2 = 1 & \text{3. Simplified} \\ \hline \end{array} \][/tex]