A) \[2\text{cosec}\theta -2\cot \theta -\theta +c\]
B) \[2\,\text{cosec}\theta -2\cot \theta +\theta +c\]
C) \[2\,\text{cosec}\theta +2\cot \theta -\theta +c\]
D) None of these
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{\frac{\text{cosec}\theta -\cot \theta }{\text{cosec}\theta +\cot \theta }\,d\theta }=\int_{{}}^{{}}{{{(\text{cosec}\theta -\cot \theta )}^{2}}d\theta }\] \[=\int_{{}}^{{}}{\text{cose}{{\text{c}}^{2}}\theta \,d\theta }+\int_{{}}^{{}}{{{\cot }^{2}}\theta \,d\theta }-2\int_{{}}^{{}}{\text{cosec}\theta \cot \theta \,d\theta }\] \[=\int_{{}}^{{}}{(2\text{cose}{{\text{c}}^{2}}\theta -1)\,d\theta }-2\int_{{}}^{{}}{\text{cosec}\theta \cot \theta \,d\theta }\] \[=2\text{cosec}\theta -2\cot \theta -\theta +c.\]
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