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If |{:(1,cos alpha, cos beta),(cos alpha...

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`

A

`1`

B

`2`

C

`3//2`

D

`1//2`

Text Solution

Verified by Experts

The correct Answer is:
A
`(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`
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Knowledge Check

  • If sin alpha + sin beta + sin gamma =3, then the value of (cos alpha + cos beta+ cos gamma) is-

    A
    1
    B
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  • If sin alpha = sin beta and cos alpha = cos beta then-

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    `sin ""(alpha + beta)/(2)=0`
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    `cos"" (alpha -beta)/(2)=0`
    C
    `cos ""(alpha+beta)/(2) =0`
    D
    `sin ""(alpha -beta)/(2)=0`

  • sin alpha + sin beta = a "and" cos alpha + cos beta = b cos (alpha + beta) =

    A
    `(a^(2)-b^(2))/(a^(2)+b^(2))`
    B
    `-(a^(2) - b^(2))/(a^(2) + b^(2))`
    C
    `(a^(2) + b^(2))/(a^(2) - b^(2))`
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