A) \[\frac{1}{2}\]
B) \[\frac{3}{4}\]
C) \[\frac{5}{4}\]
D) \[2\]
Correct Answer: B
Solution :
Given, \[\tan \theta =\frac{1}{\sqrt{7}}\,\,\,\,\Rightarrow \,\,\,\cot \,\theta =\sqrt{7}\] Now, \[\frac{(\text{cose}{{\text{c}}^{2}}\theta -{{\sec }^{2}}\theta )}{(\text{cose}{{\text{c}}^{2}}\theta +{{\sec }^{2}}\theta )}=\frac{(1+{{\cot }^{2}}\theta -1-{{\tan }^{2}}\theta )}{(1+{{\cot }^{2}}\theta +1+{{\tan }^{2}}\theta )}\] \[=\frac{{{\cot }^{2}}\,\theta -{{\tan }^{2}}\theta }{2+{{\cot }^{2}}\theta +{{\tan }^{2}}\theta }\] \[=\frac{{{(\sqrt{7})}^{2}}-{{\left( \frac{1}{\sqrt{7}} \right)}^{2}}}{2+{{(\sqrt{7})}^{2}}+{{\left( \frac{1}{\sqrt{7}} \right)}^{2}}}\] \[=\frac{49-1}{7}\times \frac{7}{63+1}\] \[=\frac{48}{64}\] \[=\frac{3}{4}\]
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