A question about hyperbolic functions Suppose $(x,y,z),(a,b,c)$ satisfy $$x^2+y^2-z^2=-1, z\ge 1,$$ $$ax+by-cz=0,$$ $$a^2+b^2-c^2=1.$$ Does it follow that $$z\cosh(t)+c\sinh(t)\ge 1$$ for all real number $t$?

Write
$$a=\rho\cos\phi,\quad b=\rho\sin\phi;\qquad x=r\cos\psi,\quad y=r\sin\psi.$$
Then $1+r^2=z^2$, whence 
$$r=\sinh \tau,\quad z=\cosh\tau$$
for some $\tau\geq0$. Similarly $c^2=a^2+b^2-1=\rho^2-1$, whence
$$\rho=\cosh\alpha,\quad c=\sinh\alpha$$
for some $\alpha\in{\mathbb R}$. Therefore we get
$$\eqalign{ax+by-cz&=\rho r(\cos\phi\cos\psi+\sin\phi\sin\psi)-\cosh\tau\sinh\alpha \cr
&=\sinh\tau\cosh\alpha\cos(\phi-\psi)-\cosh\tau\sinh\alpha\ .\cr}$$
As $ax+by-cz=0$ this implies
$$\left|{\sinh\alpha\over\cosh\alpha}\right|\leq\left|{\sinh\tau\over\cosh\tau}\right|$$
or $|\alpha|\leq\tau$.
It follows that for any real $t$ one has
$$\eqalign{z\cosh t+ c\sinh t&=\cosh\tau\cosh t+\sinh\alpha\sinh t\geq \cosh\alpha\cosh t+\sinh\alpha\sinh t\cr &=\cosh(\alpha+t)\geq1\ .\cr}$$
