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In triangle A B C ,2a csin(1/2(A-B+C)) i...

In triangle `A B C ,2a csin(1/2(A-B+C))` is equal to `a^2+b^2-c^2` (b) `c^2+a^2-b^2` (c) `b^2-c^2-a^2` (d) `c^2-a^2-b^2`

A

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

B

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

C

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

D

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

Text Solution

Verified by Experts

The correct Answer is:
B
We know that `A + B + C = 180^(@)`
or `A + C - B = 180^(@) - 2B`
Now, `2ac sin [(1)/(2) (A - B + C)]`
`= 2 ac sin (90^(@) - B) = 2ac cos B`
`= (2ac (a^(2) + c^(2) - b^(2)))/(2ac) = a^(2) + c^(2) - b^(2)`
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