If \(\alpha, \beta\) and \(\gamma\) are angles that satisfy the following conditions, find the value of \(x…
If \(\alpha, \beta\) and \(\gamma\) are angles that satisfy the following conditions, find the value of \(x y z\).
1. \(\tan (\alpha)+\tan (\beta)+\tan (\gamma)\)
\(=\tan (\alpha) \cdot \tan (\beta) \cdot \tan (\gamma)\)
2. \(x=\cos (\alpha)+i \sin (\alpha)\)
3. \(y=\cos (\boldsymbol{\beta})+i \sin (\boldsymbol{\beta})\)
4. \(z=\cos (\gamma)+i \sin (\gamma)\)
1, but not -1
-1, but not 1
\pm 1
0
Solution
It is given that,
\(\begin{aligned}
& \tan \alpha+\tan \beta+\tan \gamma=\tan \alpha \tan \beta \tan \gamma \\
& \Rightarrow \tan \alpha+\tan \beta=-\tan \gamma(1-\tan \alpha \tan \beta) \\
& \Rightarrow \quad \frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta}=\tan (-\gamma) \\
& \Rightarrow \tan (\alpha+\beta)=\tan (-\gamma) \Rightarrow \alpha+\beta=n \pi-\gamma, n \in \mathbf{I} \\
& \Rightarrow \quad \alpha+\beta+\gamma=n \pi, n \in \mathbf{I} \\
& \text {and } x=\cos \alpha+i \sin \alpha, y=\cos \beta+i \sin \beta, \\
& z=\cos \gamma+i \sin \gamma \\
& \therefore \quad x y z=e^{i \alpha} . e^{i \beta} . e^{i \gamma} \\
& \quad=e^{i(\alpha+\beta+\gamma)}=e^{i n \pi}=\cos n \pi=1 \text { or }-1 \\
& \therefore \quad x y z= \pm 1
\end{aligned}\)
Hence, option (c) is correct.