If `|{:(1,cos alpha, cos beta),(cos alpha, 1 , cos gamma ),(cos beta, cos gamma , 1):}|=|{:(0,cos alpha, cos beta),(cos alpha , 0 , cos gamma),(cos beta, cos gamma, 0):}|` then the value of `cos^2 alpha + cos^2 beta + cos^2 gamma` is : (a) `1` (b) `1/2` (c) `3/8` (d) `9/4`

Options

A. `1`
B. `2`
C. `3//2`
D. `1//2`

Answer

A

Explanation

`(a)` `|{:(1,cos alpha, cos beta),(cos alpha, 1,cosgamma),(cosbeta,cosgamma,1):}|=|{:(0,cosalpha,cosbeta),(cosalpha,0,cosbeta),(cosbeta,cosgamma,0):}|`
`impliessin^(2)gamma-cosalpha(cosalpha-cosbetacosgamma)+cosbeta(cosalphacosgamma-cosbeta)`
`=-cosalpha(-cosbetacosgamma)+cosbeta(cosalphacosgamma)`
`impliessin^(2)gamma-cos^(2)alpha+2cosalphacosbetacosgamma-cos^(2)beta=2cosalphacosbetacosgamma`
`impliessin^(2)gamma=cos^(2)alpha+cos^(2)beta`
`impliescos^(2)alpha=cos^(2)beta+cos^(2)gamma=1`