The product of all possible values of \(\alpha\), for which \(\lim _{x\to 0}\left(\frac{1-\cos (\alpha x)\cos ((\alpha +…

Using \(\cos \theta=1-\frac{\theta^2}{2}+\ldots\)

\(\cos (\alpha x) \cos ((\alpha+1) x) \cos ((\alpha+2) x) =1-\frac{\alpha^2 x^2+(\alpha+1)^2 x^2+(\alpha+2)^2 x^2}{2}+\ldots\)

So, \(1-\cos (\alpha x) \cos ((\alpha+1) x) \cos ((\alpha+2) x) =\frac{\alpha^2+(\alpha+1)^2+(\alpha+2)^2}{2} x^2\)

Also, \(\sin ^2((\alpha+1) x)=(\alpha+1)^2 x^2\)

\( \therefore \frac{1}{2(\alpha+1)^2}\left[\alpha^2+(\alpha+1)^2+(\alpha+2)^2\right]=2\)
\(\alpha^2+(\alpha+1)^2+(\alpha+2)^2=4(\alpha+1)^2 \)
\( \alpha^2+(\alpha+2)^2=3(\alpha+1)^2 \)
\( \Rightarrow \alpha^2+2 \alpha-1=0\)
Product \(=-1\)