| The value of \[\frac{1}{\text{cosec}\theta -\text{cot}\theta }-\frac{1}{\sin \theta },\]is [SSC (CGL) 2013] |
A) \[\cot \theta \]
B) \[\text{cosec}\theta \]
C) \[\tan \theta \]
D) \[1\]
Correct Answer: A
Solution :
| \[\frac{1}{\text{cosec}\theta -\text{cot}\theta }-\frac{1}{\sin \theta }\] |
| \[=\frac{1}{\frac{1}{\sin \theta }-\frac{\cos \theta }{\sin \theta }}-\frac{1}{\sin \theta }=\frac{\sin \theta }{1-\cos \theta }-\frac{1}{\sin \theta }\] |
| \[=\frac{{{\sin }^{2}}\theta -(1-\cos \theta )}{(1-\cos \theta )\sin \theta }\] |
| \[=\frac{1-{{\cos }^{2}}\theta -1+\cos \theta }{(1-\cos \theta )\sin \theta }\]\[[\because {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1]\] |
| \[=\frac{\cos \theta \,\,(1-\cos \theta )}{(1-\cos \theta )\sin \theta }=\frac{\cos \theta }{\sin \theta }=\cot \theta \] |
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