Question

Please help, Simmons, page 579 problem 4.

Prove that r1=theta and r2=1/theta intersect orthogonally at theta=1.

Pt 1: they intersect, r = 1 = 1/1 which is true.

Pt 2: Define psi1 as the angle between the line from the origin to a point on r=theta and the tangent to that point.

tan(psi1) = r1/(dr1/dtheta) = theta/1 = theta.
tan(psi2) = -theta

orthonogal implies tan(psi1-psi2) = 1
(since psi2 is negative)

this becomes:
(tan(psi1)-tan(psi2))/(1+tan(psi1)tan(psi2)) = 1
2theta/(1-theta^2) = 1

using theta=1,
2/0 = 1, which is false what am i doing wrong here?