Question

There are 4 positive integers a,b,c,d, such that 4cos(x)cos(2x)cos(4x) = cos(ax)+cos(bx)+cos(cx)+cos(dx) for all values of x. Find a+b+c+d.?

Answer 1

5

Explanation:

`4 cos(x)cos (2x)cos(4x)\\=2 cosx (2 cos (4x)cos (2x))\\=2 cos x (cos (4x+2x)/(2)+cos (4x-2x)/(2))\\=2 cos x(cos 3x+cos x)\\=2cos 3xcos x+2 cos ^2x\\=cos( (3x+x)/(2))+cos (3x-x)/(2)+cos 2x+1\\=cos2x+cos x+cos 2x+cos 0x\\a=2,b=1,c=2,d=0\\a+b+c+d=2+1+2+0=5`