<s>[INST] Find the equation in spherical coordinates of $x^2 + y^2 – z^2 = 4$. Find the equation in spherical coordinates of $x^2 + y^2 – z^2 = 4$.
$$\begin{align}
x^2 + y^2 &= r^2\sin^2(\theta)\\
z^2 &= r^2 \cos(\theta) \\
x^2 + y^2 - z^2&=r^2(\sin^2(\theta) - \cos^2(\theta)) = 4
\end{align}$$
Thus, $$-r^2(\cos(2\theta)) = 4$$
Is this right?
 [/INST] $x^2 +y^2-z^2=4 \implies \rho ^2-2z^2=4$$
$$ \implies \rho ^2(1-2\cos ^2 ( \phi ))=4$$
$$\implies \rho ^2(\cos 2( \phi ))=4 \implies \rho ^2=4\sec (2\phi)$$
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