Question

State the whether given statement is true or false

${\cos }^2(\beta -\gamma )+{\cos }^2(\gamma -\alpha )+{\cos }^2(\alpha -\beta )=1+2\cos (\beta -\alpha )\cos (\gamma -\alpha )\cos (\alpha -\beta )$.

Answer 1

${\cos }^2(\beta -\gamma )+{\cos }^2(\gamma -\alpha )+{\cos }^2(\alpha -\beta )$
$=\frac{\cos 2(\beta -\gamma )+1}{2}+\frac{\cos 2(\gamma -\alpha )+1}{2}+{\cos }^2(\alpha -\beta )$
$=\frac{1}{2}(\cos 2(\beta -\gamma )+\cos 2(y-\alpha )+2{\cos }^2(\alpha -\beta )+2)$
$=\frac{1}{2}\left(2\cos (\alpha +\beta )\cos (\alpha -\beta )+2{\cos }^2(\alpha -\beta )+2\right)$
$=\frac{1}{2}\left(2\cos (\alpha -\beta )\left(\cos (\alpha +\beta )+\cos (\alpha -\beta )\right)+2\right)$
$=\frac{1}{2}\left(2\cos (\alpha -\beta )\left(2\cos (\beta -\alpha )\cos (\gamma -\alpha )\right)+2\right)=1+2\cos (\alpha -\beta )\cos (\gamma -\alpha )\cos (\beta -\alpha )$