Question

Let `alpha, beta` be such that `pi lt alph-betalt3piif sin alpha+sinbeta=-21/65and cos alpha+cos beta =-27/65,` then the value of
`cos""(alpha-beta)/(2),` is
A. `-(6)/(65)`
B. `(3)/(sqrt130)`
C. `6/65`
D. `-(3)/(sqrt130)`

Answer 1

Correct Answer - D
We have,
`sin alpha+sinbeta=-21/65and cos alpha+cosbeta=-27/65`
`implies2sin""(alpha+beta)/(2)cos""(alpha-beta)/(2)=-21/65`
`and 2cos""(alpha+beta)/(2)cos""(alpha-beta)/(2)=-27/65`
`implies4(sin^(2)""(alpha+beta)/(2)+cos^(2)""(alpha+beta)/(2))cos^(2)""(alpha-beta)/(2)=(-(21)/(65))^(2)+(-(27)/(65))^(2)`
`implies4cos^(2)((alpha-beta)/(2))=(1170)/(65^(2))`
`implies4cos^(2)((alpha-beta)/(2))=(1170)/(4xx65^(2))`
`implies|cos""(alpha-beta)/(2)|=(sqrt(1170))/(130)`
`impliescos""(alpha-beta)/(2)=-(sqrt(1170))/(130)" "[{:(,because(pi)/(2)lt(alpha-beta)/(2)lt(3pi)/(2)),(,thereforecos""(alpha-beta)/(2)lt0):}]`
`impliescos""(alpha-beta)/(2)=-sqrt((1170)/(130xx130))=-(3)/(sqrt(130))`