Answer
$(a \times b) .[(b \times c) \times (c \times a)]=[a \cdot (b\times c)] ^2$
Work Step by Step
Use property $a \times (b \times c)=(a.c)b-(a.b)c$
$(a \times b) .[(b \times c) \times (c \times a)]=(a \times b) .[((b \times c)\cdot a)c-((b \times c)c)a]$
Since, the vectors $(b \times c)$ and $c$ are perpendicular, thus, $((b \times c)c)a]=0$
$(a \times b) .[(b \times c) \times (c \times a)]=(a \times b) .[((b \times c)\cdot a)c]$
$(a \times b) .[(b \times c) \times (c \times a)]=(a \times b) .[(a(b \times c)) c]$
$(a \times b) .[(b \times c) \times (c \times a)]=(a (b\times c)) \cdot (a (b\times c)) $
Hence, $(a \times b) .[(b \times c) \times (c \times a)]=[a \cdot (b\times c)] ^2$
