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If A+B+C=pi, prove that sin^2A/2+sin^2B/...

If `A+B+C=pi,` prove that `sin^2A/2+sin^2B/2-sin^2C/2=1-2cosA/2cosB/2sinC/2dot`

Text Solution

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