multiply up and down (numeratr and deno) by sin (x/2^n)

Pn = 2 sin (x/2^n) cos x/2^n................ cos x/2^n   /  2 sin x/2^n

..........  on doing like this u will get Pn = 1/2^nsin x cosec x/2^n

 

now we need a little bit of modification of wat is given

     tan x = cot x - 2cot x

1/2 tan 2x = 2 cot 2x - 2 cot 2^2x

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.1/2^n-1 tan 2^n-1 x = 2^n-1( cot 2^n-1 x - 2 cot 2^n x)

 

 

now u find out the limit and u will get the ans as 1/x - cot x

Beautiful problem man