Let `alphaandbeta` be such that `piltalpha-betalt3pi`. If `sinalpha+sinbeta=-(21)/(65)and cosalpha+cosbeta=-(27)/(65)`, then the value of `"cos"((alpha-beta))/(2)` is

Options

A. `-(3)/(sqrt(130))`
B. `(6)/(sqrt(130))`
C. `(6)/(65)`
D. `-(6)/(65)`

Answer

A

Explanation

Given , `sinalpha+sinbeta=(-21)/(65)` ...(i)
and `cosalpha+cosbeta=-(27)/(65)` …(ii)
on squaring and adding Eqs . (i) and (ii) , we get
`(sinalpha+sinbeta)^(2)+(cosalpha+cosbeta)^(2)=(-(21)/(65))^(2)+(-(27)/(65))^(2)`
`rArrsin^(2)alpha+sin^(2)beta+2sinalphasinbeta+cos^(2)alpha+cos^(2)beta+2cosalphacosbeta=(441)/(4225)+(729)/(4225)`
`rArr2+2cos(alpha-beta)=(1770)/(4225)`
`rArr2[2cos^(2)((alpha-beta)/(2))]=(1170)/(4225)`
`rArrcos^(2)((alpha-beta)/(2))=(1170)/(4xx4225)=(9)/(130)`
`rArrcos((alpha-beta)/(2))=-(3)/(sqrt(130)) " "[becausepiltalpha-betalt3pi]`