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If A+B+C=180^@, then prove thatcos^2 A +...

If `A+B+C=180^@`, then prove that`cos^2 A + cos^2 B +cos^2 C=1-2cosA cosB cosC`.

Text Solution

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`cos^(2)A+cos^(2)B-cos^(2)C=cos^(2)A+sin^(2)C-sin^(2)B`
`=cos^(2)A+sin(C+B)sin(C-B)`
`=1-sin^(2)A+sinA sin(C-B)`
`=1-sinA[sinA-sin(C-B)]`
`=1-sinA[sin(B+C)-sin(C-B)]`
`=1-2sin A sin B cos C`
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