In any `DeltaABC`, prove that `(a-b)^(2)cos^(2)""C/2+(a+b)^(2)sin^(2)""C/2=c^(2).`

In any `DeltaABC`, prove that `(a-b)^(2)cos^(2)""C/2+(a+b)^(2)sin^(2)"

1.	In any `DeltaABC`, prove that `(a-b)^(2)cos^(2)""C/2+(a+b)^(2)sin^(2)""C/2=c^(2).`
Answer» By the projection rule in `DeltaABC` `a=c cos B+b cos C` `b=a cos C+cos A` `thereforea+b=ccos B +b cos A ` `(a+b)(1-cos C)=c (cos B +cosA)` `(a+b) (2 sin ^(2)((C)/(2)))=c (cos B+cosA)` `(a+b)*sin^(2)((C)/(2))=c/2(cos B +cos A ) (a+b)" "...(1)` and `a-b=c cos B+b cos C-a cos C-c cos A` `(a-b)(1+cos C)=c (cos B -cos A) ` `(a-b)(2cos ^(2)((C)/(2)))=c (cos B -cos A)` `(a-b)^(2)cos ^(2)((C)/(2))=C/2(cos B-cos A)(a-b)" "...(2)` `(a-b)^(2)cos ^(2)((C)/(2))+(a+b)^(2)sin^(2)((C)/(2))=c/2[a cos B+a cos A+b cos B+` `b cos A+a cos B-a cos A-b cos B +b cos A]` lt brgt `therefore (a-b)^(2)cos ^(2)((C)/(2))+(a+b)^(2)sin^(2)((c)/(2))` `=C/2.2(a cos B+cos A)` `therefore(a-b)cos ^(2)((C)/(2))+(a+b)^(2)sin^(2)((C)/(2))=c^(2)` `[becausec=a cos B+b cos A` Hence Provide.