In `DeltaABC,(a-b)^(2)"cos"^(2)(C)/(2)+(a+b)^(2)"sin"^(2)(C)/(2)` is equal to

Options

A. `b^2`
B. `c^2`
C. `a^2`
D. `a^2+b^2+c^2`

Answer

B

Explanation

`(a-b)^(2)cos^(2)""(C)/(2)+(a+b)^(2)sin^(2)""(C)/(2)`
`=(a^(2)+b^(2)-2ab)cos^(2)""(C)/(2)+a^(2)+b^(2)+2ab sin^(2)""(C)/(2)`
`=a^(2) (cos^(2)""(C)/(2)+sin^(2)""(C)/(2))+b^(2)(cos^(2)""(C)/(2)+sin^(2)""(C)/(2))-2ab (cos^(2)""(C)/(2)-sin^(2)""(C)/(2))`
`=a^(2)+b^(2)-2ab cos C" "[therefore cos^(2)""(C)/(2)-sin^(2)""(C)/(2)=cosC]`
`=a^(2)+b^(2)-2ab ((a^(2)+b^(2)-c^(2))/(2ab))=c^(2)" "[therefore cos C=(a^(2)+b^(2))/(2ab))`