A) \[\frac{8}{1-{{x}^{8}}}\]
B) \[\frac{8}{{{x}^{8}}-1}\]
C) \[\frac{6}{{{x}^{6}}-1}\]
D) \[\frac{6}{1-{{x}^{6}}}\]
Correct Answer: B
Solution :
\[\frac{1}{x-1}-\frac{1}{x+1}-\frac{2}{{{x}^{2}}+1}-\frac{4}{{{x}^{4}}+1}\] \[=\left[ \frac{1}{x-1}-\frac{1}{x+1} \right]-\frac{2}{{{x}^{2}}+1}-\frac{4}{{{x}^{4}}+1}\] \[=\frac{(x+1)-(x-1)}{{{x}^{2}}-1}-\frac{2}{{{x}^{2}}+1}-\frac{4}{{{x}^{4}}+1}\] \[=\frac{2}{{{x}^{2}}-1}-\frac{2}{{{x}^{2}}+1}-\frac{4}{{{x}^{4}}+1}\] \[=\left[ \frac{2}{{{x}^{2}}-1}-\frac{2}{{{x}^{2}}+1} \right]-\frac{4}{{{x}^{4}}+1}\] \[=\frac{2({{x}^{2}}+1)-2({{x}^{2}}-1)}{({{x}^{2}}-1)({{x}^{2}}+1)}-\frac{4}{{{x}^{4}}+1}\] \[=\frac{2{{x}^{2}}+2-2{{x}^{2}}+2}{({{x}^{2}}-1)({{x}^{2}}+1)}-\frac{4}{{{x}^{4}}+1}\] \[=\frac{4}{{{x}^{4}}-1}-\frac{4}{{{x}^{4}}+1}\] \[=\frac{4({{x}^{4}}+1)-4({{x}^{4}}-1)}{({{x}^{4}}-1)({{x}^{4}}+1)}\] \[=\frac{4{{x}^{4}}+4-4{{x}^{4}}+4}{{{x}^{8}}-1}=\frac{8}{{{x}^{8}}-1}\]
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