Q. 75.0( 1 Vote )
For any Δ ABC show that-
2 (bc cos A + ca cos B + ab cos C) = a2 + b2 + c2
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Answer :
Note: In any ΔABC we define ‘a’ as length of side opposite to ∠A , ‘b’ as length of side opposite to ∠B and ‘c’ as length of side opposite to ∠C .
Key point to solve the problem:
Idea of cosine formula in ΔABC
• Cos A = 
• Cos B = 
• Cos C = 
As we have to prove:
2 (bc cos A + ca cos B + ab cos C) = a2 + b2 + c2
As LHS contain 2ca cos B, 2ab cos C and 2cb cos A ,which can be obtained from cosine formulae.
∴ From cosine formula we have:
Cos A = 
⇒ 2bc cos A =
…..eqn 1
Cos C = 
⇒ 2ab cos C = 
…eqn 2
And, Cos B = 
⇒ 2ac cos B = 
……eqn 3
Adding eqn 1,2 and 3:-
2bc cos A + 2ab cos C + 2ac cos B = 
+ 
⇒ 2bc cos A + 2ab cos C + 2ac cos B = 
⇒ 2(bc cos A + ab cos C + ac cos B) = 
…proved
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