A) (a) \[\cos ec\theta -\sec \theta =\cos ec\theta .\sec \theta \]
B) (b)\[\cos ec\theta .\sec \theta =1\]
C) \[\cos \theta +\sin \theta =\sqrt{2}\]
D) \[\sqrt{3}\sin \theta -\cos \theta =2\]
Correct Answer: B
Solution :
\[cosec\,\theta -sec\text{ }\theta =cosec\theta .\,sec\theta \] \[\Rightarrow \] \[\frac{\cos \theta -\sin \theta }{\cos \theta \sin \theta }=\frac{1}{\cos \theta \sin \theta }\] \[\Rightarrow \] \[\cos \theta =1+\sin \theta \] \[\therefore \]At\[\theta =0\]above equation satisfies. \[cosec\text{ }\theta .\,sec\text{ }\theta \,\text{=}\,1\] \[\Rightarrow \] \[sin\text{ }\theta \text{ }cos\text{ }\theta =1\] \[\Rightarrow \]\[2sin\text{ }\theta \text{ }cos\text{ }\theta =2\] \[\Rightarrow \]\[sin\text{ 2}\theta =2\] As we know sin \[\theta \] is not greater than 1. \[\therefore \]The above equation has no solution exist.
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