If $\alpha,\beta$ and $\gamma$ are angles such that $ an{\alpha + an{\beta + an{\gamma = an{\alpha an{\beta an\gamma}}}}}$ and $x = \cos{\alpha + i\sin{\alpha,\ \ y = \cos{\beta + i\sin\beta}}}$ and $z = \cos{\gamma + i\sin\gamma}$, then $xyz$ is equal to

Question

If $\alpha,\beta$ and $\gamma$ are angles such that $	an{\alpha + 	an{\beta + 	an{\gamma = 	an{\alpha	an{\beta	an\gamma}}}}}$ and $x = \cos{\alpha + i\sin{\alpha,\ \ y = \cos{\beta + i\sin\beta}}}$ and $z = \cos{\gamma + i\sin\gamma}$, then $xyz$ is equal to

Dr. Abubacker · Accepted Answer

(c)
Given, $	an{\alpha + 	an{\beta + 	an{\gamma = 	an{\alpha	an{\beta	an\gamma}}}}}$ …(i)
$	herefore\ \ 	an{(\alpha + \beta + \gamma)}$
$= \frac{	an{\alpha + 	an{\beta + 	an{\gamma - 	an{\alpha	an{\beta	an\gamma}}}}}}{1 - 	an{\alpha	an{\beta - 	an{\beta	an{\gamma - 	an{\gamma	an\alpha}}}}}}$
$\Rightarrow \ \ 	an{(\alpha + \beta + \gamma)}$
$= \frac{0}{1 - 	an{\alpha	an{\beta - 	an{\beta	an{\gamma - 	an{\gamma	an\alpha}}}}}}$
[From Eq. (i)]
$\Rightarrow \ 	an{(\alpha + \beta + \gamma) = 0}$
$\Rightarrow \ \ \alpha + \beta + \gamma = 0^{{^\circ}}$ or $\pi$
$	herefore xyz = \left( \cos{\alpha + i\sin\alpha} ight)\left( \cos{\beta + i\sin\beta} ight)(\cos{\gamma + i\sin\gamma})$
$= \cos{(\alpha + \beta + \gamma) + i\sin{(\alpha + \beta + \gamma)}}$
$= \cos{0{^\circ} + i\sin{0{^\circ} = 1}}$
or $xyz = \cos{\pi + i\sin{\pi = - 1}}$

Q. If $\alpha,\beta$ and $\gamma$ are angles such that $\tan{\alpha + \tan{\beta + \tan{\gamma = \tan{\alpha\tan{\beta\tan\gamma}}}}}$ and $x = \cos{\alpha + i\sin{\alpha,\ \ y = \cos{\beta + i\sin\beta}}}$ and $z = \cos{\gamma + i\sin\gamma}$, then $xyz$ is equal to

Explanation

Reviewed by Dr. Abubacker

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