If `cosalpha+cosbeta+cosgamma=0 a n d a l so sinalpha+sinbeta+singamma=0,` then prove that `cos2alpha+cos2beta+cos2gamma` `=sin2alpha+sin2beta+sin2gamma=0` `sin3alpha+sin3beta+sin3gamma=3sin(alpha+beta+gamma)` `cos3alpha+cos3beta+cos3gamma=3cos(alpha+beta+gamma)`

Options

A. 0
B. `cos(alpha+beta+gamma)`
C. `3cos(alpha+beta+gamma)`
D. `3sin(alpha+beta+gamma)`

Answer

C

Explanation

Let `a=cosalpha+isinalpha, b=cosbeta+isinbeta`,
and `c=cosgamma+isingamma`. Then,
`a+b+c=0+i0=0`
`rArr a^(3)+b^(3)+c^(3)=3abc`
`rArr (cos3alpha+isin3alpha)+(cos3beta+isin3beta)+(cos3lambda+isin3gamma)=3[cos(alpha+beta+gamma)+isin(alpha+beta+gamma)]`
`rArr cos3alpha+cos3beta+cos3gamma=3cos(alpha+beta+gamma)`.