For any Δ ABC show that -
Note: In any ΔABC we define ‘a’ as the length of the side opposite to ∠A, ‘b’ as the length of the side opposite to ∠B and ‘c’ as the length of the side opposite to ∠C.
The key point to solve the problem:
The idea of cosine formula in ΔABC
• Cos A =
• Cos B =
• Cos C =
As we have to prove:
The form required to prove contains similar terms as present in cosine formula.
∴ Cosine formula is the perfect tool for solving the problem.
As we see the expression has bc term so we will apply the formula of cosA
As cos A =
⇒ 2bc cos A = b2 + c2 – a2
We need (b + c )2 in our proof so adding 2bc both sides –
∴ 2bc + 2bc cos A = b2 + c2 +2bc – a2
⇒ 2bc ( 1 + cos A) = (b + c)2 - a2
∵ 1 + cos A = 2cos2 (A / 2) { using multiple angle formulae }
∴
⇒ …..Hence proved.