If alpha and beta are the roots of the quad | vedclass

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Question

$x^{2}+x \sin 	heta-2 \sin 	heta=0$

$\alpha+\beta=-\sin 	heta$

$\alpha \beta=-2 \sin 	heta$

$	ext { Now, } \frac{\alpha^{12}+\beta^{12}}

Vedclass · Accepted Answer

$\frac{{{2^{12}}}}{{{{\left( {\sin \,	heta  + 8} ight)}^{12}}}}$

Vedclass · Answer

$x^{2}+x \sin 	heta-2 \sin 	heta=0$

$\alpha+\beta=-\sin 	heta$

$\alpha \beta=-2 \sin 	heta$

$	ext { Now, } \frac{\alpha^{12}+\beta^{12}}{\left(\frac{1}{\alpha^{12}}+\frac{1}{\beta^{12}}ight)(\alpha-\beta)^{24}}$

$=\frac{(\alpha \beta)^{12}}{(\alpha-\beta)^{24}}$

$=\frac{(\alpha \beta)^{12}}{\left((\alpha+\beta)^{2}-4 \alpha \betaight)^{12}}$

$=\left[\frac{\alpha \beta}{(\alpha+\beta)^{2}-4 \alpha \beta}ight]^{12}$

$=\left(\frac{-2 \sin 	heta}{\sin ^{2} 	heta+8 \sin 	heta}ight)^{12}$

$=\frac{2^{12}}{(\sin 	heta+8)^{12}}$

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