`cosx*cos2x*cos4x.....cos(2^(n-1)x)=(sin2^n x)/(2^nsinx)AAn in N`

Prove that cos x+cos3x+cos5x+cos(2n-1)x=(sin2nx)/(2)sin x

If f(x) = cos x cos 2x cos 2^2 x cos^(2^3) x .....cos 2^(n-1) x and n gt 1 then f^(1)(pi/2) is

Prove that cosx+cos2x+ … + cosnx=(sin(n+1/2)x-sin(x/2))/(2sin(x/2)) and hence prove that : int_0^1(sin(n+1/2)x)/sin(x/2)dx=pi

If cos((x)/(2))cos((x)/(2^(2)))cos((x)/(2^(square)))......cos((x)/(2^(n)))=(sin x)/(sin((x)/(2^(n)))) prove (1)/(2)tan((x)/(2))+(1)/(4)tan((x)/(4))...(.1)/(2^(2n))tan((x)/(2^(n)))=(1)/(2^(n))cot((x)/(2^(n)))-cot x

If f(x) = cos x\ cos 2x\ cos 2^2\ x\ cos 2^3 x\ ....cos2^(n-1) x and n gt 1, then f'(pi/2) is

If (x)=(cos x+i sin x)(cos3x+i sin3x)...(cos(2n-1)x+i sin(2n-1)x) then f(x) is