Proving hyperbolic equality

Question

While trying to prove a problem on catenoids intersecting orthogonally with $S^2$ I have to prove that if $a=\frac{1}{T\cosh(T)}$ where $T$ is the only real number such that $T\tanh(T)=1$ and $t$ satisfies $t^2 + a^2\cosh(t/a)^2=1 \quad (1)$ then \begin{equation} \tanh(t/a)=a\cosh(t/a) \quad (2) \end{equation} and \begin{equation} t\cosh(t/a)=1 \quad (3) \end{equation} I have manged to show that $(1)+(2)\Rightarrow(3)$ so that I'm only left with proving that $(1)\Rightarrow(2)$. Through the following algebraic manipulation \begin{gather} t^2 + a^2\cosh(t/a)^2=1\\ t^4 + t^2a^2\cosh(t/a)^2=t^4+a^2=t^2 \end{gather} I got rid of the hyperbolic cosine, however other than verifying numerically that this is true I can't find any other way to prove this. I'm not sure whether there is a way to obtain an exact result but any help would be appreciated