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If A,B,C are the angles of a triangal ,p...

If A,B,C are the angles of a triangal ,prove that : cos A +cos B + cos `C = 1+r/R`

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cos A +cos B +cos C =2 cos `((A+B)/(2))cos ((A-B)/(2))+cos C`
`=2 sin"" C/2 cos ((A-B)/(2))+1-2 sin ^2""C/2=1+sin"" C/2= 1+2 sin """C/2[cos ((A-B)/(2))-sin (C/2)]`
`1+2sin""C/2[cos ((A-B)/(2))-cos((A+B)/(2))]" "{ therefore ""C/2= 90^@-((A+B)/(2))}`
`= 1+2 sin""C/(2).2 sin ""A/2.sin""B/2=1 +4 sin"" A/2.sin""B/2.sin""C/2`
`1+r/R" "{as,r = 4R sin A//2. sin B//2.sinC//2}`
`rArr cos A+cos B +cosC = 1+r/R.`
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