Question

If $\alpha +\beta =\gamma$, then ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1+2\cos \alpha \cos \beta \cos \gamma$.

Answer 1

We want $+1$ in the R.H.S.
Also $\alpha +\beta =\gamma$
L.H.S. $={\cos }^2\alpha +1{\sin }^2\beta +{\cos }^2\gamma$
$=1+({\cos }^2\alpha -{\sin }^2\beta )+{\cos }^2\gamma$
$=1+\cos (\alpha +\beta )\cos (\alpha -\beta )+{\cos }^2\gamma$
$=1+\cos \gamma \left[\cos \right(\alpha -\beta )+\cos (\alpha +\beta \left)\right]$
$[\because \gamma =\alpha +\beta ]$
$=1+\cos \gamma \left[2\cos \alpha \cos \beta \right]$
$=1+2\cos \alpha \cos \beta \cos \gamma$.