= Solution
Use the <Minkowski metric> with signature $(+,-)$, so the <Sine-Gordon equation> is $\phi_{tt}-\phi_{xx}+\sin\phi=0$. The <light-cone coordinates> here satisfy
$$
\partial_+=\partial_x+\partial_t,\qquad\partial_-=\partial_x-\partial_t,\qquad\partial_+\partial_-=\partial_x^2-\partial_t^2.
$$
In particular, there is no extra factor of four with this coordinate normalization. The <Bäcklund transformation> requires $a\ne0$, because one of its equations contains $a^{-1}$.
Put $F=(\psi+\phi)/2$ and $G=(\psi-\phi)/2$. In the parameter convention of this paper the <Sine-Gordon Bäcklund transformation> gives $G_+=a\sin F$ and $F_-=a^{-1}\sin G$. Differentiating, for a twice differentiable transformed field, gives
$$
G_{+-}=\cos F\sin G,\qquad F_{+-}=\cos G\sin F.
$$
Since $\psi=F+G$, the addition formula yields $\psi_{+-}=\sin(F+G)=\sin\psi$. Therefore \b[the transformed field satisfies the same <Sine-Gordon equation>:]
$$
\boxed{\psi_{tt}-\psi_{xx}+\sin\psi=0.}
$$
Subtracting the two differentiated equations also gives $\phi_{+-}=\sin\phi$, showing the compatibility with the seed equation. This proves the unheaded request before the numbered parts, without assuming a particular <soliton> form.
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