A) \[\cos ec\theta -\sec \theta =\cos ec\theta \,.\,\,\sec \theta \]
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 :
\[\text{cosec }\theta \text{-sec}\theta \text{=cosec}\theta \text{.sec}\theta \] \[\Rightarrow \] \[\frac{\cos \,\theta -\sin \,\theta }{\cos \theta \,sin\theta }=\frac{1}{\cos \,\theta \,\sin \theta }\] \[\Rightarrow \] \[\cos \,\theta =1+\sin \theta \] \[\therefore \] At \[\theta =0\] above equation satisfies. \[\text{cosec}\,\theta \text{.}\,\text{sec}\theta \text{=1}\] \[\Rightarrow \] \[\sin \theta \,\cos \theta =1\] \[\Rightarrow \] \[2\,\sin \theta \,\,\cos \theta =2\] \[\Rightarrow \] \[\sin \,2\theta =2\] As we know sin 9 is not greater than 1. \[\therefore \] The above equation has no solution exist.
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