Assume that \(\cfrac{a}{b} = \cfrac{b}{c} = \cfrac{c}{d} = k\) [where \(k\) is a nonzero constant].
\(\therefore a = bk, \quad b = ck, \quad c = dk\)
\(\therefore b = dk \times k = dk^2, \quad a = d k^2 \times k = dk^3\)
LHS \( = (a^2 - b^2)(c^2 - d^2)\)
\( = \{(dk^3)^2 - (dk^2)^2\} \{(dk)^2 - d^2\}\)
\( = \{d^2 k^6 - d^2 k^4\} \{d^2 k^2 - d^2\}\)
\( = d^2 k^4 (k^2 - 1) \times d^2 (k^2 - 1)\)
\( = d^4 k^4 (k^2 - 1)^2\)
RHS \( = (b^2 - c^2)^2\)
\( = \{(dk^2)^2 - (dk)^2\}^2\)
\( = \{d^2 k^4 - d^2 d^2\}^2\)
\( = \{d^2 k^2 (k^2 - 1)\}^2\)
\( = d^4 k^4 (k^2 - 1)^2\)
\(\therefore\) LHS \( = \) RHS [Proved].