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Solve the following equations:
(i) cos x+cos 2x+cos 3x=0
(ii) cos x+cos 3x-cos 2x=0
(iii) sin x+sin 5x=sin 3x
(iv) cos x cos 2x cos 3x=14
(v) cos x+sin x=cos 2x+sin 2x
(vi) sin x+sin 2x+sin 3=0
(vii) sin x+sin 2x+sin 3x+sin 4x=0
(viii) sin 3x-sin x=4 cos2 x-2
(ix) sin 2x-sin 4x+sin 6x=0

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Solution

(i) cos x + cos 2x + cos 3x = 0
Now,
(cosx + cos3x) + cos2x= 0 2 cos 4x2 cos 2x2 + cos2x= 0 2 cos2x cosx + cos2x = 0 cos2x ( 2 cosx+1) = 0
cos 2x = 0 or, 2 cos x + 1 = 0
cos 2x = cos π2 or cos x =-12= cos 2π3
2x = (2n + 1) π2, n Z or x= 2mπ ± 2π3, m Z
x= (2n + 1)π4, n Z or x= 2mπ ± 2π3, m Z

(ii) (cosx + cos3x) - cos2x = 0
2 cos 4x2 cos 2x2 - cos2x = 0 2 cos2x cosx - cos2x= 0 cos2x ( 2 cosx - 1) = 0

cos2x = 0 or 2 cosx - 1 = 0
cos2x = cos π2 or cosx= 12 cosx= cosπ3
2x = (2n + 1)π2, n Z or x= 2mπ ± π3, m Z
x = (2n + 1)π4, n Z or x= 2mπ ± π3, m Z

(iii) sinx + sin5x = sin3x
2 sin6x2 cos 4x2 = sin3x2 sin3x cos2x = sin3x2 sin3x cos2x - sin3x= 0 sin3x (2 cos2x- 1) = 0

sin3x = 0 or (2 cos2x - 1) = 0
sin3x = sin 0 or cos2x = 12 = cos π3
3x = nπ or 2x = 2mπ ± π3
x= nπ3, n Z or x= mπ ± π6, m Z

(iv) cosx cos2x cos3x= 14cosx+2x+cos2x-x2cos3x=142cos3x+cosxcos3x=12cos23x+2cosx cos3x-1=02cos23x-1+2cosx cos3x=0cos6x+cos4x+cos2x=0cos6x+cos2x+cos4x=02cos4xcos2x+cos4x=0cos4x2cos2x+1=0cos4x=0 or 2cos2x+1=0cos4x=0 or cos2x=-12cos4x=cosπ2 or cos2x=cos2π34x=2n+1π2, nZ or 2x=2mπ±2π3, mZx=2n+1π8, nZ or x=mπ±π3, mZ

(v) cosx+ sinx = cos2x + sin2x

cosx - cos2x = sin2x - sinx- 2 sin 3x2 sin -x2 = 2 sin x2 cos 3x2 2 sin 3x2 sin x2 = 2 sin x2 cos 3x2 2 sin x2 sin 3x2 - cos 3x2 = 0

sin x2 = 0 or sin 3x2 - cos 3x2 = 0
sin x2 = sin 0 or sin 3x2 = cos 3x2
x2 = nπ, n Z or cos 3x2 = cos π2 - 3x2
x = 2nπ, n Z or 3x2 = 2mπ ± π2 - 3x2, m Z
x= 2nπ, n Z or 3x2 = 2mπ + π2 - 3x2, m Z (Taking negative sign will give absurd result.)
x= 2nπ, n Z or x= 2mπ3 + π6, m Z

(vi) sinx + sin2x + sin3x = 0

sinx + sin3x+ sin2x = 0 2 sin 4x2 cos 2x2 + sin2x = 0 2 sin2xcosx + sin2x= 0 sin2x (2 cosx + 1) = 0

sin2x = 0 or 2 cosx + 1 = 0
sin2x = sin 0 or cosx=-12 cosx= cos 2π3
x= nπ2, n Z or x = 2mπ ± 2π3, m Z

(vii) sinx+ sin2x+ sin3x + sin4x = 0

sin3x + sinx + sin4x+ sin2x = 0 2 sin 4x2 cos 2x2 + 2 sin 6x2 cos 2x2 = 0 2 sin2x cosx + 2 sin3x cos x = 0 2 cosx ( sin2x+ sin3x ) = 0 2 cosx2 sin 5x2 cos x2 = 0 4 cosx sin 5x2 cos x2 = 0
cos x = 0 , sin 5x2 = 0 or cos x2 = 0
cos x= cos π2, sin 5x2 = sin 0 or cos x2 = cos π2
x= (2n + 1) π2, nZ or 5x2= nπ , n Z or, x2 = (2n+ 1) π2 , n Z
x= (2n + 1) π2 , n Z or x= 2nπ5 , n Z or x = (2n + 1)π, n Z

(viii) sin3x - sinx = 4 cos2x - 2
sin3x - sinx= 2 ( 2 cos2x - 1) 2 sin 2x2 cos 4x2 = 2 cos 2x 2 sinx cos2x = 2 cos2x sinx cos2x = cos2x cos2x ( sinx - 1) = 0
cos 2x = 0 or sinx- 1 = 0
cos 2x= cos π2 or sinx = 1sinx= sin π2
2x = (2n + 1)π2, n Z or x= nπ + (-1)n π2 , n Z
x = (2n+ 1)π4 , nZ or x= nπ + (-1)n π2 , nZ

(ix) sin 2x - sin 4x+ sin 6x = 0.
2 sin 8x2 cos 4x2 - sin4x= 0 2 sin4x cos2x - sin4x = 0 sin4x ( 2 cos2x - 1) = 0
sin 4x = 0 or 2 cos2x- 1 = 0
4x= nπ , n Z or cos2x = 12 cos2x= cos π3
x= nπ4, n Z or x= nπ ± π6, n Z

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