| If \[\text{cosec}\,\,\text{39}{}^\circ =x,\] then the value of \[\frac{1}{\text{cose}{{\text{c}}^{2}}51{}^\circ }+{{\sin }^{2}}39{}^\circ +{{\tan }^{2}}51{}^\circ -\frac{1}{{{\sin }^{2}}51{}^\circ {{\sec }^{2}}39{}^\circ }\] is [SSC (CPO) 2011] |
A) \[\sqrt{{{x}^{2}}-1}\]
B) \[\sqrt{1-{{x}^{2}}}\]
C) \[{{x}^{2}}-1\]
D) \[1-{{x}^{2}}\]
Correct Answer: C
Solution :
| Given, \[\text{cosec}\,\,\text{39}{}^\circ =x\] |
| Then,\[\frac{1}{\text{cose}{{\text{c}}^{2}}51{}^\circ }+{{\sin }^{2}}39{}^\circ +{{\tan }^{2}}51{}^\circ \] |
| \[-\frac{1}{{{\sin }^{2}}51{}^\circ \cdot {{\sec }^{2}}39{}^\circ }\] |
| \[={{\sin }^{2}}51{}^\circ +{{\sin }^{2}}39{}^\circ +{{\tan }^{2}}(90{}^\circ -39{}^\circ )\] |
| \[-\frac{1}{{{\sin }^{2}}(90{}^\circ -39{}^\circ )\cdot {{\sec }^{2}}39{}^\circ }\] |
| \[={{\cos }^{2}}39{}^\circ +{{\sin }^{2}}39{}^\circ +{{\cot }^{2}}39{}^\circ -\frac{1}{{{\cos }^{2}}39{}^\circ \cdot {{\sec }^{2}}39{}^\circ }\] \[=1+{{\cot }^{2}}39{}^\circ -1\] |
| \[=\text{cose}{{\text{c}}^{2}}39{}^\circ -1\] \[[\because 1+{{\cot }^{2}}\theta =\text{cose}{{\text{c}}^{2}}\theta ]\] |
| \[={{x}^{2}}-1\] |
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