1+cos2x+cos4x + cos six x is equal to 4 cosx cos2x cos3x
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The proof can be done starting from right side.
Recall the formula
2 cos A cos B = cos (A+B)+cos (A-B)
and also cos 2A = 2cos^2 A-1
consider the right side
= 4 cos x cos 2x cos 3x
= 2(2cosx cos 3x) cos2x
= 2cos 2x [cos 4x+cos2x]
= 2 cos 2x cos 4x +2 cos ^2 2x
= [cos 6x + cos 2x] + 2 cos^2 2x
= cos 6x + cos 2x + 1 + cos 4x
= 1+cos 2x+cos 4x + cos 6x
Hence proved.
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