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If [(cos^2 theta,cos theta sin theta),(...

If `[(cos^2 theta,cos theta sin theta),(cos theta sin theta,sin^2 theta)] [(cos^2 phi,cos phi sin phi),(cos phi sin phi,sin^2 phi)]=O` 'then value of ' `theta- phi` `equals,`

A

`2npi, in Z`

B

`n pi/2, n in Z`

C

`(2n+1) pi/2, n in Z`

D

`npi, n in Z`

Text Solution

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The correct Answer is:
C
`AB=[(cos^(2) theta,),(cos theta sin theta,)][(cos^(2) phi,cos phi sin phi),(cos phi sin phi,sin^(2) phi)]`
`=[(cos^(2) theta cos^(2) phi+cos theta cos phi sin theta sin phi ,cos^(2)theta cos phi sin phi+cos theta sin theta sin^(2) phi),(cos theta sin thete cos^(2) phi+sin^(2) theta cos phi sin phi,cos theta cos phi sin theta sin phi+sin^(2) theta sin^(2) phi)]`
`=[(cos theta cos phi (cos (theta-phi)),cos theta sin phi (cos(theta-phi))),(sin theta cos phi (cos (theta-phi)),sin theta sin phi (cos (theta-phi)))]`
`=(cos (theta-phi)) [(cos theta cos phi,cos theta sin phi),(sin theta cos phi,sin theta sin phi)]`
Now, `AB=O implies cos (theta-phi)=0`
`implies theta-phi=(2n+1) pi//2, n in Z`.
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