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In any DeltaABC, prove that (a-b)^(2)cos...

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

A

`b^(2)`

B

`c^(2)`

C

`a^(2)`

D

`a^(2)+b^(2)+c^(2)`

Text Solution

Verified by Experts

The correct Answer is:
B
`(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)xx1+b^(2)-2ab((s(s-c))/(ab)-((s-a)(s-b))/(ab))`
`=a^(2)+b^(2)-(2ab)/(ab)[((a+b+c)/(2))((a+b+c)/(2)-c)`
`-((a+b+c)/(2)-a)xx((a+b+c)/(2)-b)`
`=a^(2)+b^(2)-(2)/(4)[(a+b+c)(a+b-c)-(b+c-a)(a+c-b)]`
`=a^(2)+b^(2)-(1)/(2)[(a+b^(2))-c^(2)-(ab+bc-b^(2)+ac+c^(2)-bc-a^(2)-ac+ab)]`
`=a^(2)+b^(2)-(1)/(2)[a^(2)+b^(2)+2ab-c^(2)+a^(2)+b^(2)-c^(2)-2ab]`
`=a^(2)+b^(2)-(1)/(2)[2a^(2)+2ab^(2)-2c^(2)]`
`=a^(2)+b^(2)-[a^(2)+b^(2)-c^(2)]=c^(2)`
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