Question

If $\cos \alpha +\cos \beta +\cos \gamma +\cos (\alpha +\beta +\gamma )=m\cos \frac{\alpha +\beta }{2}\cos \frac{\beta +\gamma }{2}\cos \frac{\gamma +\alpha }{2}$.Find $m$

Answer 1

Given that

$\cos \alpha +\cos \beta +\cos \gamma +\cos \left(\alpha +\beta +\gamma \right)=m\cos \frac{\alpha +\beta }{2}\cos \frac{\beta +\gamma }{2}\cos \frac{\gamma +\alpha }{2}$

$\Rightarrow 2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}+2\cos \frac{\alpha +\beta +\gamma +\gamma }{2}\cos \frac{\alpha +\beta +\gamma -\gamma }{2}=m\cos \frac{\alpha +\beta }{2}\cos \frac{\beta +\gamma }{2}+\cos \frac{\gamma +\alpha }{2}$

$\Rightarrow 2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}+2\cos \left(\frac{\alpha +\beta +2\gamma }{2}\cos \frac{\alpha +\beta }{2}\right)=m\cos \frac{\alpha +\beta }{2}\cos \frac{\beta +\gamma }{2}\cos \frac{\gamma +\alpha }{2}$

$\Rightarrow 2\cos \frac{\alpha +\beta }{2}\left[\cos \frac{\alpha -\beta }{2}+\cos \frac{\alpha +\beta +2\gamma }{2}\right]=m\cos \frac{\alpha +\beta }{2}\cos \frac{\beta +\gamma }{2}\cos \frac{\gamma +\beta }{2}$

$\Rightarrow 2\left[\cos \frac{\alpha +\beta +2\gamma }{2}+\cos \frac{\alpha -\beta }{2}\right]=m\cos \frac{\beta +\gamma }{2}\cos \frac{\gamma +\alpha }{2}$

$\Rightarrow 2\left[2\cos \frac{\alpha +\beta +2\gamma +\alpha -\beta }{2\times 2}\cos \frac{\alpha +\beta +2\gamma -\alpha +\beta }{2\times 2}\right]=m\cos \frac{\beta +\gamma }{2}\cos \frac{\gamma +\alpha }{2}$

$\Rightarrow 2\left[2\cos \frac{2\alpha +2\gamma }{2\times 2}\cos \frac{2\beta +2\gamma }{2\times 2}\right]=m\cos \frac{\beta +\gamma }{2}\cos \frac{\gamma +\alpha }{2}$

$\Rightarrow 4\cos \frac{\alpha +\gamma }{2}\cos \frac{\beta +\gamma }{2}=m\cos \frac{\beta +\gamma }{2}\cos \frac{\gamma +\alpha }{2}$

$\Rightarrow 4=m$

Hence, we get,

$\Rightarrow m=4$