Convert the polar equation $r^{2} \cos 2 \theta=1$ to rectangular form.

Step 1 ：Given the polar equation \(r^{2} \cos 2 \theta=1\)

Step 2 ：We can convert this to rectangular form using the following equations:

Step 3 ：\(x = r \cos \theta\)

Step 4 ：\(y = r \sin \theta\)

Step 5 ：\(r^{2} = x^{2} + y^{2}\)

Step 6 ：\(\cos \theta = x / r\)

Step 7 ：\(\sin \theta = y / r\)

Step 8 ：First, we can rewrite \(r^{2}\) as \(1/\cos(2\theta)\)

Step 9 ：Then, we can convert this to rectangular form to get \(1/(x^{2} - y^{2})\)

Step 10 ：Finally, we replace \(r^{2}\) with \(x^{2} + y^{2}\) in the equation

Step 11 ：So, the rectangular form of the given polar equation is \(\boxed{\frac{1}{x^{2} - y^{2}} = x^{2} + y^{2}}\)