A) \[{{x}^{2}}{{y}^{2}}{{z}^{2}}\]
B) \[-{{x}^{2}}{{y}^{2}}{{z}^{2}}\]
C) 1
D) None of these
Correct Answer: B
Solution :
| \[\frac{\left[ \frac{x}{y}-\frac{y}{x} \right]\left[ \frac{y}{z}-\frac{z}{y} \right]\left[ \frac{z}{x}-\frac{x}{z} \right]}{\left[ \frac{1}{{{x}^{2}}}-\frac{1}{{{y}^{2}}} \right]\left[ \frac{1}{{{y}^{2}}}-\frac{1}{{{z}^{2}}} \right]\left[ \frac{1}{{{z}^{2}}}-\frac{1}{{{x}^{2}}} \right]}\] |
| \[=\frac{\left[ \frac{{{x}^{2}}-{{y}^{2}}}{xy} \right]\left[ \frac{{{y}^{2}}-{{z}^{2}}}{zy} \right]\left[ \frac{{{z}^{2}}-{{x}^{2}}}{xz} \right]}{\left[ \frac{{{y}^{2}}-{{x}^{2}}}{{{x}^{2}}{{y}^{2}}} \right]\left[ \frac{{{z}^{2}}-{{y}^{2}}}{{{z}^{2}}{{y}^{2}}} \right]\left[ \frac{{{x}^{2}}-{{z}^{2}}}{{{x}^{2}}{{z}^{2}}} \right]}\] |
| \[=-\frac{\frac{({{x}^{2}}-{{y}^{2}})({{y}^{2}}-{{z}^{2}})({{z}^{2}}-{{x}^{2}})}{{{x}^{2}}{{y}^{2}}{{z}^{2}}}}{\frac{({{x}^{2}}-{{y}^{2}})({{y}^{2}}-{{z}^{2}})({{z}^{2}}-{{x}^{2}})}{{{x}^{4}}{{y}^{4}}{{z}^{4}}}}\] |
| \[=-\frac{{{x}^{4}}{{y}^{4}}{{z}^{4}}}{{{x}^{2}}{{y}^{2}}{{z}^{2}}}\] |
| \[=-\,\,{{x}^{2}}{{y}^{2}}{{z}^{2}}\] |
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