Convert the polar equation $r^{2} \cos 2 \theta=1$ to rectangular form.
So, the rectangular form of the given polar equation is \(\boxed{\frac{1}{x^{2} - y^{2}} = x^{2} + y^{2}}\)
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 \(\boxed{\frac{1}{x^{2} - y^{2}} = x^{2} + y^{2}}\)