Find a polar equation of the form r2=f(θ) for the curve represented by the cartesian equation x2−y2=1. (note the power of r in the equation! )

The equations to transform from Cartesian to polar :
x = r cos θ   , y = r sin θ
Or
x² = r² cos² θ   , y² = r² sin² θ

To represent the Cartesian equation x²−y²=1
substitute with the values of x² and y²

∴ r² cos² θ - r² sin² θ = 1
   r² (cos² θ - sin² θ) = 1
but cos² θ - sin² θ = cos 2θ
∴ r² cos 2θ = 1
   r² = 1/cos 2θ

∴ r² = sec 2θ  ⇒⇒⇒⇒ The required equation