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In ∆ ABC, Prove that: a2 = (b + c)2 − 4bc \(cos^2\frac{A}{2}\).

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RHS = (b + c)2 − 4bc cos2\(\frac{A}{2}\).

= b2 + c2 + 2bc − 4bc \((\frac{cos\,A + 1}{2})\)

= b2 + c2 + 2bc – 2bc (cos A + 1)

= b2 + c2 − 2bc .\(\frac{b^2+c^2-a^2}{2bc}\)

= b2 + c2 – (b2 + c2 – a2)

= b2 + c2 – b2 – c2 + a2

= a2 = LHS

∴ LHS = RHS

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