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In DeltaABC, prove that: (b-c)^(2)cos^...

In `DeltaABC`, prove that:
`(b-c)^(2)cos^(2)A/2+(b+c)^(2)sin^(2)A/2=a^(2)`

Text Solution

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LHS `=(b-c)^(2)cos^(2)A/2+(b+c)^(2)sin^(2)A/2`
`=(b^(2)+c^(2)-2bc)cos^(2)A/2+(b^(2)+c^(2)+2bc)sin^(2)A/2`
`=b^(2)(cos^(2)A/2+sin^(2)A/2)+c^(2)(cos^(2)A/2+sin^(2)A/2)-2bc(cos^(2)A/2-sin^(2)A/2)`
`=b^(2)+c^(2)-2bccosA`
`=b^(2)+c^(2)-2bc.(b^(2)+c^(2)-a^(2))/(2bc)`
`=b^(2)+c^(2)-(b^(2)+c^(2)-a^(2))`
`=a^(2)`= RHS Hence Proved.
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