If `A+B=pi/3 and cos A+cos B=1,` then which of the following is/are true ?

Options

A. `cos(A-B)=1/3`
B. `|cosA-cosB|=sqrt(2/3)`
C. `cos(A-B)=-2/3`
D. `|cosA-cosB|=(1)/(2sqrt3)`

Answer

B, C

Explanation

We have,
`cosA+cos B=1`
`implies2 cos ""(A+B)/(2)cos""(A-B)/(2)=1`
`impliescos""(A-B)/(2)=(1)/(sqrt(3))" "[becauseA+B=(pi)/(3)]`
`implies2 cos^(2)""((A-B)/(2))-1=2/3-1impliescos(A-B)=-1/3`
Now, `|cosA-cosB|`
`=|2sin((A+B)/(2))sin((B-A)/(2))|`
`=|2sin""(pi)/(6)sin((A-B)/(2))|`
`= |sin((A-B)/(2))|=sqrt(1-cos^(2)((A-B)/(2)))=sqrt(1-(1)/(3))=sqrt((2)/(3))`
So, option (b) is also true,