Take LHS:
\[\left[ 1\text{ }\text{ }cos\text{ }2x\text{ }+\text{ }sin\text{ }2x \right]\text{ }/\text{ }\left[ 1\text{ }+\text{ }cos\text{ }2x\text{ }+\text{ }sin\text{ }2x \right]\]
From formulae, \[cos\text{ }2x\text{ }=\text{ }1\text{ }\text{ }2\text{ }si{{n}^{2}}~x\]
\[=\text{ }2\text{ }co{{s}^{2}}~x\text{ }\text{ }1\]
\[Sin\text{ }2x\text{ }=\text{ }2\text{ }sin\text{ }x\text{ }cos\text{ }x\]
SO,