Question

State the whether given statement is true or false

If $A+B=C$,prove that ${\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C=2\cos A\cos B\cos C$.

• A. True
• B. False

Answer 1

The correct option is B False

${\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C=\frac{\cos 2A+1}{2}+\frac{\cos 2B+1}{2}+\frac{\cos 2C+1}{2}$

$=\frac{1}{2}(\cos 2A+\cos 2B+\cos 2C+3)$
$=\frac{1}{2}\left(2\cos (A+B)\cos (A-B)+\cos (A+B)+3\right)$
$=\frac{1}{2}\left(2\cos (A+B)\cos (A-B)+2{\cos }^2(A+B)-1+3\right)$
$=\frac{1}{2}\left(2\cos (A+B)\left(\cos (A-B)+\cos (A+B)\right)+2\right)=\cos C\left(2\cos A\cos B\right)+2=2\cos A\cos B\cos C+2$