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In Δ ABC with the usual notations prove that `(a-b)^2 cos^2(C/2)+(a+b)^2sin^2(C/2)=c^2`

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#### Solution

LHS= `(a-b)^2 cos^2(C/2)+(a+b)^2sin^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-a^2-b^2+c^2`

`=c^2`

=RHS

Hence proved

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