a) K (representing kappa or curvature)=|r'×r''|/|r'|³ by definition.
The cross-product of the two vectors is given by the determinant:
| i j k|
| f' g' 0 | =
| f'' g'' 0 |
<0,0,f'g''-g'f''> which has magnitude |f'g''-g'f''|.
r'=<f',g',0> so |r'|=√((f')²+(g')²).
Therefore K=|f'g''-g'f''|/((f')²+(g')²)^(3/2).
b) f=tan⁻¹(sinh(t)), g=ln(cosh(t)),
f'=cosh(t)/(1+sinh²(t))=cosh(t)/cosh²(t)=sech(t),
f''=-sech(t)tanh(t);
g'=sinh(t)/cosh(t)=tanh(t),
g''=sech²(t).
|f'g''-g'f''|=|sech³(t)+sech(t)tanh²(t)|.
(f')²+(g')²=sech²(t)+tanh²(t)=1
Note: cosh²(t)-sinh²(t)=1,
1-tanh²(t)=sech²(t), hence sech²(t)+tanh²(t)=1.
Therefore K=|sech³(t)+sech(t)tanh²(t)|=|sech(t)|.
When t=ln(5), sinh(t)=(5-⅕)/2=12/5, cosh(t)=(5+⅕)/2=13/5,
tanh(t)=12/13, sech(t)=5/13.
Curvature K=5/13.