Take LHS:
\[si{{n}^{2}}~\left( \pi /8\text{ }+\text{ }x/2 \right)\text{ }\text{ }si{{n}^{2}}~\left( \pi /8\text{ }\text{ }x/2 \right)\]
From formula, \[\text{ }si{{n}^{2}}~A\text{ }\text{ }si{{n}^{2}}~B\text{ }=\text{ }sin\text{ }\left( A+B \right)\text{ }sin\text{ }\left( A-B \right)\]
SO,
\[si{{n}^{2}}~\left( \pi /8\text{ }+\text{ }x/2 \right)\text{ }\text{ }si{{n}^{2}}~\left( \pi /8\text{ }\text{ }x/2 \right)\text{ }=\text{ }sin\text{ }\left( \pi /8\text{ }+\text{ }x/2\text{ }+\text{ }\pi /8\text{ }\text{ }x/2 \right)\text{ }sin\text{ }\left( \pi /8\text{ }+\text{ }x/2\text{ }\text{ }\left( \pi /8\text{ }\text{ }x/2 \right) \right)\]
\[=\text{ }sin\text{ }\left( \pi /8\text{ }+\text{ }\pi /8 \right)\text{ }sin\text{ }\left( \pi /8\text{ }+\text{ }x/2\text{ }\text{ }\pi /8\text{ }+\text{ }x/2 \right)\]
\[=\text{ }sin\text{ }\pi /4\text{ }sin\text{ }x\]
\[=\text{ }1/\surd 2\text{ }sin\text{ }x\text{ }\left[ since,\text{ }since\text{ }\pi /4\text{ }=\text{ }1/\surd 2 \right]\]
= RHS
Hence proved.