Question

In triangle $ABC$, $a(b^2+c^2)\cos A+b(c^2+a^2)\cos B+c(a^2+b^2)\cos C=$

• A. $2a^2{b}^2{c}^2$
• B. $3a^2{b}^2{c}^2$
• C. $3abc$
• D. $2abc$

Answer 1

The correct option is C $3abc$

$a(b^2+c^2)\cos A+b(c^2+a^2)\cos B+c(a^2+b^2)\cos C$
$=a{b}^2\cos A+a{c}^2\cos A+b{c}^2\cos B+b{a}^2\cos B+c{a}^2\cos C+c{b}^2\cos C$
$=ab(b\cos A+a\cos B)+bc(b\cos C+c\cos B)+ac(c\cos A+a\cos C)$
$=abc+abc+abc=3abc$[Using projection formula]