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

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To rewrite the equation $$ r^{2} \cos(2\theta) = 25 $$ in terms of $$ \sin(\theta) $$ and $$ \cos(\theta) $$, we can use the double angle identity for cosine:

$$
\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)
$$

We also know that in polar coordinates, $$ r^2 = x^2 + y^2 $$, where $$ x = r \cos(\theta) $$ and $$ y = r \sin(\theta) $$.

Using the identity for $$ \cos(2\theta) $$, we can substitute it into the original equation:

$$
r^2 \cos(2\theta) = r^2 (\cos^2(\theta) - \sin^2(\theta)) = 25
$$

Now, substituting $$ r^2 = x^2 + y^2 $$ and using $$ \cos(\theta) = \frac{x}{r} $$ and $$ \sin(\theta) = \frac{y}{r} $$, we can express $$ \cos^2(\theta) $$ and $$ \sin^2(\theta) $$ in terms of $$ x $$ and $$ y $$:

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

Thus, we can rewrite $$ \cos(2\theta) $$:

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

Substituting this back into the equation gives:

$$
r^2 \left(\frac{x^2 - y^2}{r^2}\right) = 25
$$

This simplifies to:

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

Thus, the equation of the curve in terms of $$ \sin(\theta) $$ and $$ \cos(\theta) $$ is:

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

This is the final form of the equation.
The equation of the curve in terms of $$ \sin(\theta) $$ and $$ \cos(\theta) $$ is $$ x^2 - y^2 = 25 $$.

Consider the following curve.

Write an equation for the curve in terms of and

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