Question

Cos 3x + cos 4x + cos 5x + cos 6x = ....

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Answer 1

Verified answer

Jawab

cos 3x + cos 4x + cos 5x + cos 6x =
= (cos 6x + cos 4x)  + (cos 5x + cos 3x)
= 2 cos ¹/₂(6x+4x)  cos ¹/₂(6x-4x) + 2 cos ¹/₂(5x+3x) cos ¹/₂(5x-3x)
= 2 cos 5x  cos x + 2 cos 4x cos x
= 2 cos x ( cos 5x + cos 4x)
= 2 cos x ( cos 4x + cos (4x +x))
= 2 cos x (cos 4x + cos 4x sin x - sin 4x cos x)
= 2 cos x { cos 4x( 1 + sin x)  - cos x (sin 4x)}
= 2 cos x {(8cos⁴ x - 8cos²x -1)(1 +sin x)  - cos x (4 sin x cos x -8 sin³ x cos x)}
=2 cos x ( 8cos⁴x + 8cos⁴x sin x -8cos²x-8cos²xsin x -1 - sin x - 4 sinx cos²x +8 sin³x cos² x)
= 16 cos⁵ x + 16cos⁵x sin x -16 cos³x -16cos³x sinx - 2cos x - 2sinx cosx -8 cos³x sin x + 16 sin³x cos³ x
= 16 cos⁵ x (1+sin x)  - 8cos³ x(2 + 2sinx + sin x + 2 sin³ x) -2cos x(1-sinx)
= 16 cos⁵ x (1+ sin x) - 8cos³ x( 2+ 3 sin x + 2 sin³ x) - 2cos x(1 -sin x)