Question : Find the value of $\frac{\cos 37^{\circ}}{\sin 53^{\circ}}-\cos 47^{\circ} \operatorname{cosec} 43^{\circ}$.
Option 1: 0
Option 2: –1
Option 3: 2
Option 4: 1
Correct Answer: 0
Solution :
$\frac{\cos 37^{\circ}}{\sin 53^{\circ}}-\cos 47^{\circ} \operatorname{cosec} 43^{\circ}$
= $\frac{\cos (90-53)^{\circ}}{\sin 53^{\circ}}-\cos (90-43)^{\circ} \operatorname{cosec} 43^{\circ}$ [$\sin \theta = \cos(90^\circ - \theta)$]
= $\frac{\sin 53^{\circ}}{\sin 53^{\circ}}-\sin 43^{\circ} \operatorname{cosec} 43^{\circ}$
= 1 – 1 [$\because \sin \theta = \frac{1}{\operatorname{cosec} \theta}$]
= 0
Hence, the correct answer is 0.
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