| If\[\frac{\cos x}{\text{cosec}\,\,x+1}+\frac{\cos x}{\text{cosec}\,\,x-1}=2,\]which one of the following is one of the value of x? |
A) \[\frac{\pi }{2}\]
B) \[\frac{\pi }{3}\]
C) \[\frac{\pi }{4}\]
D) \[\frac{\pi }{6}\]
Correct Answer: C
Solution :
| \[\frac{\cos x}{\text{cosec}\,\,x+1}+\frac{\cos x}{\text{cosec}\,\,x-1}=2\] |
| \[\Rightarrow \]\[\frac{\cos x\,\,(\text{cosec}\,\,x-1)+\cos x(\text{cosec}\,\,x+1)}{(\text{cosec}\,\,x+1)(\text{cosec}\,\,x-1)}\] |
| \[\Rightarrow \]\[\frac{\cos x\,\,\text{cosec}\,\,x-\cos x+\text{cos }x\,\,\text{cosec}\,\,x+\cos x}{\text{cose}{{\text{c}}^{2}}\,\,x-1}=2\] |
| \[\Rightarrow \]\[\frac{2\cos x\,\,\text{cosec}\,\,x}{{{\cot }^{2}}x}=2\]\[\Rightarrow \]\[\frac{2\cot x}{{{\cot }^{2}}x}=2\] |
| \[\Rightarrow \]\[\cot x=1\]\[\Rightarrow \]\[x=\frac{\pi }{4}\] |
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