Question

Say true or false:

If $A+B+C=\pi$, then $\cos 2A+\cos 2B+\cos 2C+4\cos A\cos B\cos C$ is positive.

• A. True
• B. False

Answer 1

The correct option is B False

As $A+B+C=\pi$

$\cos 2A+\cos 2B+\cos 2C$

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

$=2\cos (\pi -C)\cos (A-B)+2{\cos }^{2}C-1$ since $\cos C+\cos D=2\cos \frac{C+D}{2}\cos \frac{C-D}{2},\cos 2x=2{\cos }^{2}x-1$

$=-2\cos C(\cos (A-B)+\cos (A+B))-1$

$=-2\cos C(\cos (A-B)+\cos (A+B))-1$

$=-4\cos A\cos B\cos C-1$

$\cos 2A+\cos 2B+\cos 2C=-4\cos A\cos B\cos C-1$

Hence $\cos 2A+\cos 2B+\cos 2C+4\cos A\cos B\cos C=-1<0$.