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In any DeltaABC, prove that Delta=(a^(2)...

In any `DeltaABC`, prove that `Delta=(a^(2)-b^(2))/2(sinAsinB)/(sin(A-B))`.

A

`2Delta`

B

`4Delta`

C

`Delta`

D

`3Delta`

Text Solution

Verified by Experts

`((a^(2)-b^(2))sinAsinB)/(2sin(A-B))`
`=((asinB)(asinA)-(bsinA)(bsinB))/(2sin(A-B))`
`=(asinB(asinA-bsinB))/(2sin(A-B))` `[becauseasinB=bsinA]`
`=(kasinB(sin^(2A-sin^(2)B))/(2sin(A-B))` `[because a/(sinA)=b/(sinB)+c/(sinC)=k]`
`=(kiasinBsin(A+B)sin(A-B))/(2sin(A-B))`
`=(kasinBsinC)/2=1/2absinC` `[becauseksinB=b]`
`=Delta`
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