Question #2678

How to do this proving question:
cosx+cos3x+cos5x+cos7x=4cos0.5xcos2.5xcosx.

Expert's answer

Use the trigonometric identity of the sum of cosines:& cos a + cos b = 2 cos (a+b)/2& cos (a-b)/2.

cos x + cos 3x + cos 5x + cos 7x = 2 cos(x+3x)/2 cos(x-3x)/2 & + 2 cos(5x + 7x)/2 cos( 5x - 7x)/2 =

= 2 cos 2x cos x + 2 cos 6x& cos x = 2 cosx (cos 2x + cos 6x) = 4 cos x * cos (2x + 6x)/2& * cos(2x - 6x)/2 =

= 4 cos x * cos 4x * cos 2x

cos x + cos 3x + cos 5x + cos 7x = 2 cos(x+3x)/2 cos(x-3x)/2 & + 2 cos(5x + 7x)/2 cos( 5x - 7x)/2 =

= 2 cos 2x cos x + 2 cos 6x& cos x = 2 cosx (cos 2x + cos 6x) = 4 cos x * cos (2x + 6x)/2& * cos(2x - 6x)/2 =

= 4 cos x * cos 4x * cos 2x

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