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If the sum of all the solutions of the equation `8 cosx.(cos(pi/6+x)cos(pi/6-x)-1/2)=1` in `[0,pi]` is `k pi` then k is equal to

A

`20//9`

B

`2//3`

C

`13//9`

D

`8//9`

Text Solution

Verified by Experts

The correct Answer is:
C
`8 cos x (cos (pi/6+x) cos (pi/6-x)-1/2)=1`
`rArr 8 cos x [cos^(2) pi/6- sin^(2) x-1/2]=1`
`rArr 8 cos x [1/4-(1-cos^(2) x)]=1`
`rArr 8cos^(3)x-6 cos x =1`
`rArr 4 cos^(3) x-3 cos x=1/2`
`rArr cos 3 x=1/2="cos" pi/3`
`3x=2npi pm pi/3, n in Z`
Since `x in [0, pi]`, possible values are
`3x=pi/3, 2pi pm pi/3`
`:. x=pi/9, (5pi)/9, (7pi)/9`
sum of solution `=(13 pi)/9`
`:. k=13/9`
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