Take LHS:

\[3{{\left( sin\text{ }x\text{ }\text{ }cos\text{ }x \right)}^{~4}}~+\text{ }6\text{ }{{\left( sin\text{ }x\text{ }+\text{ }cos\text{ }x \right)}^{~2}}~+\text{ }4\text{ }(si{{n}^{6}}~x\text{ }+\text{ }co{{s}^{6}}~x)\]

From formula,

\[{{\left( a\text{ }+\text{ }b \right)}^{2}}~=\text{ }{{a}^{2}}~+\text{ }{{b}^{2}}~+\text{ }2ab\]

\[{{\left( a\text{ }\text{ }b \right)}^{2}}~=\text{ }{{a}^{2}}~+\text{ }{{b}^{2}}~\text{ }2ab\]

\[{{a}^{3}}~+\text{ }{{b}^{3}}~=\text{ }\left( a\text{ }+\text{ }b \right)\text{ }({{a}^{2}}~+\text{ }{{b}^{2}}~\text{ }ab)\]

So,

\[3{{\left( sin\text{ }x\text{ }\text{ }cos\text{ }x \right)}^{~4}}~+\text{ }6\text{ }{{\left( sin\text{ }x\text{ }+\text{ }cos\text{ }x \right)}^{~2}}~+\text{ }4\text{ }(si{{n}^{6}}~x\text{ }+\text{ }co{{s}^{6}}~x)\text{ }=\text{ }3{{\{{{\left( sin\text{ }x\text{ }\text{ }cos\text{ }x \right)}^{~2}}\}}^{2}}~+\text{ }6\text{ }\{{{\left( sin\text{ }x \right)}^{2}}~+\text{ }{{\left( cos\text{ }x \right)}^{2}}~+\text{ }2\text{ }sin\text{ }x\text{ }cos\text{ }x)\}\text{ }+\text{ }4\text{ }\{{{(si{{n}^{2}}~x)}^{3}}~+\text{ }{{(co{{s}^{2}}~x)}^{3}}\}\]\[=\text{ }3\{{{\left( sin\text{ }x \right)}^{~2}}~+\text{ }{{\left( cos\text{ }x \right)}^{2}}~\text{ }2\text{ }sin\text{ }x\text{ }cos\text{ }x){{\}}^{2}}~+\text{ }6\text{ }(si{{n}^{2}}~x\text{ }+\text{ }co{{s}^{2}}~x\text{ }+\text{ }2\text{ }sin\text{ }x\text{ }cos\text{ }x)\text{ }+\text{ }4\{(si{{n}^{2}}~x\text{ }+\text{ }co{{s}^{2}}~x)\text{ }(si{{n}^{4}}~x\text{ }+\text{ }co{{s}^{4}}~x\text{ }\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x)\}\]

\[=\text{ }3\left( 1\text{ }\text{ }2\text{ }sin\text{ }x\text{ }cos\text{ }x \right){{~}^{2}}~+\text{ }6\text{ }\left( 1\text{ }+\text{ }2\text{ }sin\text{ }x\text{ }cos\text{ }x \right)\text{ }+\text{ }4\{\left( 1 \right)\text{ }(si{{n}^{4}}~x\text{ }+\text{ }co{{s}^{4}}~x\text{ }\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x)\}\]

We know, \[si{{n}^{2}}~x\text{ }+\text{ }co{{s}^{2}}~x\text{ }=\text{ }1\]

So,

\[=\text{ }3\{{{1}^{2}}~+\text{ }{{\left( 2\text{ }sin\text{ }x\text{ }cos\text{ }x \right)}^{~2}}~\text{ }4\text{ }sin\text{ }x\text{ }cos\text{ }x\}\text{ }+\text{ }6\text{ }\left( 1\text{ }+\text{ }2\text{ }sin\text{ }x\text{ }cos\text{ }x \right)\text{ }+\text{ }4\{{{(si{{n}^{2}}~x)}^{2}}~+\text{ }{{(co{{s}^{2}}~x)}^{2}}~+\text{ }2\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x\text{ }\text{ }3\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x)\}\]

\[=\text{ }3\{1\text{ }+\text{ }4\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x\text{ }\text{ }4\text{ }sin\text{ }x\text{ }cos\text{ }x\}\text{ }+\text{ }6\text{ }\left( 1\text{ }+\text{ }2\text{ }sin\text{ }x\text{ }cos\text{ }x \right)\text{ }+\text{ }4\{{{(si{{n}^{2}}~x\text{ }+\text{ }co{{s}^{2}}~x)}^{~2}}~\text{ }3\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x)\}\]

\[=\text{ }3\text{ }+\text{ }12\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x\text{ }\text{ }12\text{ }sin\text{ }x\text{ }cos\text{ }x\text{ }+\text{ }6\text{ }+\text{ }12\text{ }sin\text{ }x\text{ }cos\text{ }x\text{ }+\text{ }4\{{{\left( 1 \right)}^{2}}~\text{ }3\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x)\}\]

\[=\text{ }9\text{ }+\text{ }12\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x\text{ }+\text{ }4(1\text{ }\text{ }3\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x)\]

\[=\text{ }9\text{ }+\text{ }12\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x\text{ }+\text{ }4\text{ }\text{ }12\text{ }si{{n}^{2}}~x\text{ }co{{s}^{2}}~x\]

\[=13\]

= RHS

Hence proved.