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If A+B+C+D = 2pi, prove that : cosA +cos...

If `A+B+C+D = 2pi`, prove that : `cosA +cosB+cosC+cosD=4 cos, (A+B)/2 cos, (B+C)/(2) cos, (C+A)/2`

A

`4cos.(A+B)/(2)cos.(B+C)/(2)cos.(C+A)/(2)`

B

`4sin.(A+B)/(2)sin.(B+C)/(2)sin.(C+A)/(2)`

C

`1-4sin.(A+B)/(2)sin.(B+C)/(2)sin.(C+A)/(2)`

D

`-1-4cos.(A+B)/(2)cos.(B+C)/(2)cos.(C+A)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
A
`L.H.S.=(cos A+ cos B)+(cos C + cos D)`
`=2 cos.(A+B)/(2)cos.(A-B)/(2)+2 cos.(C+D)/(2)cos.(C-D)/(2)`
`=2cos.(A+B)/(2)cos.(A-B)/(2)cos.(A-B)/(2)+2cos(pi-(A+B)/(2))cos.(C-D)/(2)`
`=2 cos.(A+B)/(2)cos.(A-B)/(2)-2cos.(A+B)/(2)cos.(C-D)/(2)`
`=2 cos.(A+B)/(2)[cos.(A-B)/(2)-cos.(C-D)/(2)]`
`=2 cos.(A+B)/(2)xx 2 sin(A-B+C-D)/(4)sin.(C-D-A+B)/(4)`
`=4 cos.(A+B)/(2)sin.(A+C-(B+D))/(4)sin.(B+C-(A+D))/(4)`
`=4 cos.(A+B)/(2)sin.(A+C-(2pi-A-C))/(4)sin.(B+C-(2pi-B-C))/(4)`
`=4 cos.(A+B)/(2)sin.((A+C)/(2)-(pi)/(2))sin((B+C)/(2)-(pi)/(2))`
`=4 cos.(A+B)/(2)[-sin((pi)/(2)-(A+C)/(2))][-sin((pi)/(2)-(B+C)/(2))]`
`=4 cos.(A+B)/(2)cos.(A+C)/(2)cos.(B+C)/(2)`
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