The angle between corresponding tangent lines on two Bertrand curves is constant and torsions of the two associate Bertrand curves have the same sign and their product is constant
Two distinct parametrized curves $\boldsymbol{x}$ and $\boldsymbol{y}$ are called Bertrand mates if for each $t$, the normal line to $\boldsymbol{x}$ at $\boldsymbol{x}(t)$ equals the normal line to $\boldsymbol{y}$ at $\boldsymbol{y}(t)$.
So, we have $$\boldsymbol{y}(s_1)=\boldsymbol{x}(s)+a(s)\boldsymbol{n}(s)$$ Now we have to calculate $\boldsymbol T_x$ and $\boldsymbol T_y$:
$$ \begin{align} \boldsymbol T_y&=\boldsymbol y'(s_1)\\ &=\boldsymbol x'(s)+a'(s)\boldsymbol n(s)+a(s)\boldsymbol n'(s)\\ &= \boldsymbol x'(s)+a'(s)\boldsymbol n(s)+a(s)(-\kappa \boldsymbol t(s)+\tau\boldsymbol b(s))\\ &=\boldsymbol x'(s)-\kappa a(s)\boldsymbol t(s)+a'(s)\boldsymbol n(s)+\tau a(s)\boldsymbol b(s)\\\\ \boldsymbol T_x&=\boldsymbol x'(s) \end{align} $$ Now, consider, $\boldsymbol T_x . \boldsymbol T_y=\|\boldsymbol T_x\|\|\boldsymbol T_y\|\cos\theta$. But from this, I couldn't see the angle $\theta$ is constant. Any help will be appreciated.