Dear,
cosx/{[2x/(2n+1)

] +1/2 } can be written as
2(2n+1)

cosx/[4x+(2n+1)

]
now [4x+(2n+1)

]=t
you will get the form cos({t/4}-A)/t where A=(2n+1)

/4
expand the cos as cos(C+D) = cosC.cosD-sinCsinD
and i think now you can solve the further , ask if doubt any.