Consider the following curve. $$ r^{2} \cos (2 \theta)=25 $$ Write an equation for the curve in terms of $$ \sin (\theta) $$ and $$ \cos (\theta) $$ Find a Cartesian equation for the curve.

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To convert the polar equation $$ r^{2} \cos(2\theta) = 25 $$ into Cartesian coordinates, we can use the following relationships between polar and Cartesian coordinates:

$$
x = r \cos(\theta)
$$
$$
y = r \sin(\theta)
$$
$$
r^2 = x^2 + y^2
$$
$$
\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = \frac{x^2 - y^2}{r^2}
$$

Now, we can rewrite the equation $$ r^{2} \cos(2\theta) = 25 $$ using these relationships.

First, substitute $$ \cos(2\theta) $$:

$$
r^2 \cos(2\theta) = r^2 \left( \frac{x^2 - y^2}{r^2} \right) = x^2 - y^2
$$

Thus, the equation becomes:

$$
x^2 - y^2 = 25
$$

This is the Cartesian equation for the curve.

In summary, the Cartesian equation for the curve given by $$ r^{2} \cos(2\theta) = 25 $$ is:

$$
x^2 - y^2 = 25
$$
The Cartesian equation for the curve is $$ x^2 - y^2 = 25 $$.

Consider the following curve.

Write an equation for the curve in terms of and
Find a Cartesian equation for the curve.

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Consider the following curve. Write an equation for the curve in terms of and Find a Cartesian equation for the curve.