| The value of \[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\] is equal to |
A) \[\cos ec\text{ }x+\cot \text{ }x\]
B) \[\cos ec\text{ }x+\tan \text{ }x\]
C) \[\sec \text{ }x+\tan \text{ }x\]
D) \[\cos ec\text{ }x-\cot \text{ }x\]
Correct Answer: A
Solution :
| \[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\times \frac{(\sqrt{1+\sin x}+\sqrt{1-\sin x})}{(\sqrt{1-\sin x}+\sqrt{1-\sin x})}\] |
| \[\Rightarrow \] \[\frac{{{(\sqrt{1+\sin x}+\sqrt{1-\sin x})}^{2}}}{1+\sin x-(1-\sin x)}\] |
| \[\Rightarrow \] \[\frac{1+\sin x+1-\sin x+2\,(\sqrt{1+\sin x}.\sqrt{1-\sin x}}{2\sin x}\] |
| \[\Rightarrow \] \[\frac{2+2\,(\sqrt{1+\sin x}.\sqrt{1-\sin x}}{2\sin x}\] |
| \[\Rightarrow \] \[\cos ecx+\frac{2\,(\sqrt{1-{{\sin }^{2}}x}}{2\sin x}\] |
| \[\Rightarrow \] \[\cos ec\,x+\frac{\cos x}{\sin x}=\cos ec\,x+\cot x\] |
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