Question

If $A+B+C=\pi$, then prove that $\cos 2A+\cos 2B+\cos 2C=-1-4\cos A\cos B\cos C$

Answer 1

Given: $A+B+C=\pi$

$L.H.S=\cos 2A+\cos 2B+\cos 2C$

$=\left[2\cos \right(A+B\left)\cos \right(A-B\left)\right]+(2{\cos }^{2}C-1)$........$\cos C+\cos D=2\cos \frac{C+D}{2}\cos \frac{C-D}{2}$

$=2\cos (180-C)\cos (A-B)+2{\cos }^{2}C-1$

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

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

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

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