Which of the following sets can be the subset of the general solution of `1+c a s3x=2cos2x(n in Z)?` `npi+pi/3` (b) `npi+pi/6` `npi-pi/6` (d) `2npi`

Options

A. `n pi+pi/3`
B. `n pi+pi/6`
C. `n pi- pi/6`
D. `2 n pi`

Answer

B, C, D

Explanation

`1+cos 3x=2 cos 2x`
or `1+4 cos^(2) x-3 cos x=2(2 cos^(2) x-1)`
or `4 cos^(3) x-4 cos^(2) x-3 cos x+3=0`
Let `cos x=y`, we have
`4y^(3)-4y^(2)-3y+3=0`
or `4y^(2) (y-1)-3(y-1)=0`
or `(y-1) (4y^(2)-3)=0`
`rArr y=1 or y^(2)=3/4`
`rArr cos x=1 or cos^(2) x=3/4`
`rArr cos x =1 or cos^(2) x= cos^(2) pi//6`
`rArr x=2 n pi or x= n pi pm (pi//6), n in Z`