A) \[\frac{{{2}^{6}}}{{{\left( \sin \theta +8 \right)}^{12}}}\]
B) \[\frac{{{2}^{12}}}{{{\left( \sin \theta -8 \right)}^{6}}}\]
C) \[\frac{{{2}^{12}}}{{{\left( \sin \theta -4 \right)}^{12}}}\]
D) \[\frac{{{2}^{12}}}{{{\left( \sin \theta +8 \right)}^{12}}}\]
Correct Answer: D
Solution :
\[\frac{{{\alpha }^{12}}+{{\beta }^{12}}}{\left( \frac{1}{{{\alpha }^{12}}}+\frac{1}{{{\beta }^{12}}} \right){{\left( \alpha -\beta \right)}^{24}}}=\frac{{{\left( \alpha \beta \right)}^{12}}}{{{\left( \alpha -\beta \right)}^{24}}}\] \[=\frac{{{\left( \alpha \beta \right)}^{12}}}{{{\left[ {{\left( \alpha +\beta \right)}^{2}}-4\alpha \beta \right]}^{12}}}={{\left[ \frac{\alpha \beta }{{{\left( \alpha +\beta \right)}^{2}}-4\alpha \beta } \right]}^{12}}\] \[={{\left( \frac{-2\sin \theta }{{{\sin }^{2}}\theta +8\sin \theta } \right)}^{12}}=\frac{{{2}^{12}}}{{{\left( \sin \theta +8 \right)}^{12}}}\]
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