Question

In any $\Delta$ ABC
2 [bc cosA + ca cosB + ab cosC] =

• A. $a^2+b^2+c^2$
• B. $a^2-b^2+c^2$
• C. $a^2+b^2-c^2$
• D. $a^2-b^2-c^2$

Answer 1

The correct option is A $a^2+b^2+c^2$

$△ABC$

$\begin{array}{c}2[bc\cos A+ca\cos B+ab\cos C]=\\ \text{(A)}{a}^2+b^2+c^2\\ \text{(B)}{a}^2-b^2+c^2\\ \text{(c)}{a}^2+b^2-c^2\\ \text{(D)}{a}^2+b^2-c^2\end{array}$

Solution

$\begin{array}{c}=2[bc\cos A+\mathrm{cacos}B+ab\cos c]\\ =2[bc\left(\frac{b^2+c^2-a^2}{2bc}\right)+ca\left(\frac{a^2+c^2-b^2}{ac}\right)+ab\left(\frac{a^2+b^2-c^2}{ab}\right)]\\ =b^2+c^2-a^2+a^2+c^2-b^2+a^2+b^2-c^2\\ =a^2+b^2+c^2\end{array}$

Hence, (A) is the correct option.