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In any A B C , prove that: 2{asin^2C/2+...

In any ` A B C` , prove that: `2{asin^2C/2+csin^2A/2}=a+c-b`

Text Solution

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LHS `=asin^(2)C/2+csin^(2)A/2`
`=a.((s-a)(s-b)/(ab))+c.((s-b)(s-c))/(bc)`
`=(s-b)/b(s-a+s-c)=(2s-2b)/(2b)(2s-a-c)`
`=a.s/a[(s-c)/b-(s-b)/c]`
`=s[(cs-c^(2)-bs+b^(2))/(bc)]`
`=s/(bc)[(b^(2)-c^(2))-bs+cs]`
`=s/bc[(b-c)(b+c)-(b-c).s]`
`=s/(bc).(b-c)(b+c-s)`
`=(b-c)(s(2s-a-s))/(bc)`
`=(b-c)(s(s-a))/(bc)`
`=(b-c).cos^(2)A/2` = RHS Hence Proved.
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