It is an easy question. The answer is 64/36:

one way:

replace 1-cos 8x and 1-cos 6x by 2sin²4x and 2sin²3x respectively and multiply and divide by appropriate values.

I get answer as 16/9

second way:

we know lim x>0 1-cos(2x) / x²   is 1/2

dividing and multiplying by appropriate values I got 64/36

Now the question is why are the 2 answers different??

I know I am doing a stupid mistake somewhere but WHERE

64/36 and 16/9 are the same.there is no mistake

I must be going crazy.  Its time to get some rest - I have been solving IIT and AIEEE questionson limits for the last 4-5hrs.

@ spideyunlimited:

multiply and divide by 16x² to sin²⁴x and multiply and divide 9x² to sin²3x.