| What is the value of \[\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }?\] |
A) \[2\,\,\text{coses}\,\,\theta \]
B) \[2\sec \theta \]
C) \[\sec \theta \]
D) \[\text{cosec}\theta \]
Correct Answer: A
Solution :
| Let \[\text{f}\,(\theta )=\frac{\sin \theta }{1+\cos \theta }+\frac{1+\cos \theta }{\sin \theta }\] |
| \[=\frac{{{\sin }^{2}}\theta +{{(1+\cos \theta )}^{2}}}{\sin \theta \,\,(1+\cos \theta )}\] |
| \[=\frac{{{\sin }^{2}}\theta +1+{{\cos }^{2}}\theta +2\cos \theta }{\sin \theta \,\,(1+\cos \theta )}\] |
| \[=\frac{2+2\cos \theta }{\sin \theta \,\,(1+\cos \theta )}\] |
| \[=\frac{2\,\,(1+\cos \theta )}{\sin \theta \,\,(1+\cos \theta )}\] |
| \[=2\,\,\text{cosec}\theta \] |
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