Question

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

Options

• `13/9`

• `8/9`

• `20/9`

• `2/3`

Answer 1

If sum of all the solutions of the equation `8cosx.(cos(π/6 + x).cos(π/6 - x) - 1/2)` = 1 in [0, π] is kπ, then k is equal to `underlinebb(13/9)`.

Explanation:

`8cosx[cos(π/6 + x)cos(π/6 - x) - 1/2]` = 1

Using 2 cos A cos B = cos(A + B) + cos(A − B), we get

`8cosx[(cos(π/3) + cos(2x))/2 - 1/2]` = 1

∴ `4cosx[1/2 + cos(2x) - 1]` = 1

∴ `4cosx[cos(2x) - 1/2]` = 1

∴ 4 cosx cos(2x) – 2 cos x = 1

using 2 cos A cos B = cos(A + B) + cos(A − B), we get

2(cos 3x + cos x) − 2 cos x = 1

∴ cos 3x = `1/2`

∴ 3x = `2"n"π ± π/3`

∴ x = `(2"n"π)/3 ± π/9`

Solutions in [0, π] are  `π/9, (2π)/3 - π/9, (2π)/3 + π/9`

Hence, their sum = `π/9 + (5π)/9 + (7π)/9 = (13π)/9`

∴ k = `13/9`

This is the required solution.