The value of `cos12^@+cos84^@+cos156^@+cos132^@` is

Options

A. `(1)/(2)`
B. 1
C. `-(1)/(2)`
D. `(1)/(8)`

Answer

C

Explanation

Given expression, `cos12^(@)+cos84^(@)+cos150^(@)+cos132^(@)`
`" "=cos12^(@)+cos150^(@)+cos84^(@)+cos132^(@)`
` " " =2cos ((12^(@)+150^(@))/(2))*cos((12^(@)-150^(@))/(2))+2cos((84^(@)+132^(@))/(2))*cos((84^(@)-132^(@))/(2))`
`" "=2cos84^(@)cos72^(@)+2cos108^(@)*cos24^(@)`
`" "=2cos84^(@)cos(90^(@)-18^(@))+2cos(90^(@)+18^(@))*cos24^(@)`
`" "=2cos84^(@)sin18^(@)-2sin18^(@)*cos24^(@)`
`" "=2sin18^(@)(cos84^(@)-cos24^(@))`
`" "=2sin18^(@)*2sin((84^(@)+24^(@))/(2))*sin((84^(@)-24^(@))/(2))`
`" "=-4sin18^(@)*sin54^(@)sin30^(@)`
`" "=-4((sqrt(5)-1)/(4))*cos36^(@)*(1)/(2)`
`" "=-(sqrt(5)-1)((sqrt(5)+1)/(4))*(1)/(2)=-((5-1)/(8))=(-4)/(8)=(-1)/(2)`