1)

If sum of all the solutions of the equation

8cosx.(cos(π6+x).cos(π6x)12)=1 in [0,π ] is kπ , then k is equal to


A) 23

B) 139

C) 89

D) 209

Answer:

Option B

Explanation:

Key Idea Apply the  identity 

cos(x+y)cos(xy)=cos2xsin2y and   cos3x=4cos3x3cosx

We have,

8cosx(cos(π6+x)cos(π6x)12)=1

8cosx(cos2π2sin2x12)=1

  8cosx(34sin2x12)=1

  8cosx(34121+cos2x)=1

  8cosx(3+4cos2x4)=1

2(4cos3x3cosx)=1

2cos3x=1cos3x=12

  3x=π3,5π3,7π3

   [0≤ 3x ≤ 3π ]

   x=π9,5π9,7π9

Sum  =π9+5π9+7π9=13π9kπ=13π9

Hence    k= 139