A) \[\text{cosec}\,\theta -\cot \theta \]
B) \[\text{cosec}\,\theta +\cot \theta \]
C) \[\text{cose}{{\text{c}}^{2}}\,\theta +{{\cot }^{2}}\theta \]
D) \[\text{cose}{{\text{c}}^{2}}\,\theta -{{\cot }^{2}}\theta \]
Correct Answer: A
Solution :
| [a] We have, \[\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}=\sqrt{\frac{1-\cos \theta }{1+\cos \theta }\times \frac{1-\cos \theta }{1-\cos \theta }}\] |
| \[=\sqrt{\frac{{{(1-\cos \theta )}^{2}}}{1-{{\cos }^{2}}\theta }}\]\[[(a+b)\,\,(a-b)={{a}^{2}}-{{b}^{2}}]\] |
| \[=\sqrt{\frac{{{(1-\cos \theta )}^{2}}}{{{\sin }^{2}}\theta }}\]\[[1-{{\cos }^{2}}\theta ={{\sin }^{2}}\theta ]\] |
| \[=\frac{1-\cos \theta }{\sin \theta }=\frac{1}{\sin \theta }-\frac{\cos \theta }{\sin \theta }=\cos ec\,\theta -\cot \theta \] |
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