`Delta ABC` में सिद्ध कीजिए कि - `2(a "sin"^(2)(C )/(2)+c "sin"^(2)(A)/(2))=c+a-b`.

Explanation

`L.H.S. = ("a sin"^(2)(C )/(2)+"c sin"^(-2)(A)/(2))`
`=a.2 sin^(2)C//2 + c.2 sin^(2)A//2`
`= a(1-cos C)+c.(1-cos A) [because sin^(2)theta//2 = 1-cos theta]`
`= a-a ((a^(2)+b^(2)-c^(2))/(2ab))+c - c ((b^(2)+c^(2)-a^(2))/(2bc))`
`= a+c-((a^(2)+b^(2)-c^(2))/(2b))-((b^(2)+c^(2)-a^(2))/(2b))`
`=a+c-((a^(2)+b^(2)-c^(2)+b^(2)+c^(2)-a^(2))/(2b))`
`= a+c-(2b^(2))/(2b)=a+c-b = R.H.S. " "` इति सिद्धम्