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If A+B+C=pi, then cos^(2)A+cos^(2)B+cos^...

If `A+B+C=pi`, then `cos^(2)A+cos^(2)B+cos^(2)C` is equal to

A

1-cosAcosBcosC

B

1-2sinAsinBsinC

C

1-sinAsinBsinC

D

1-2cosAcosBcosC

Text Solution

Verified by Experts

The correct Answer is:
D
We have, `cos^(2)A+cos^(2)B+cos^(2)C`
`=cos^(2)A+(1-sin^(2)B)+cos^(2)C`
`=(cos^(2)A-sin^(2)B)+cos^(2)C+1`
`=cos(A+B)cos(A-B)+cos^(2)C+1`
`=cos(pi-C)cos(A-B)+cos^(2)C+1`
`=-cosCcos(A-B)+cos^(2)C+1`
`=-cosC[cos(A-B)-cosC]+1`
`=-cosC[cos(A-B)+cos(A+B)]+1`
`=-cosC(2cosAcosB)+1`
`=1-2cosAcosBcosC`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-FACTORIZATION FORMULAE -EXERCISE 2
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