A) \[\cot \theta /2\]
B) \[\tan \theta /2\]
C) \[\frac{1}{2}\cos e{{c}^{2}}\frac{\theta }{2}\]
D) \[-\frac{1}{2}\cos e{{c}^{2}}\frac{\theta }{2}\]
Correct Answer: A
Solution :
Here \[x=a\,(\theta -\sin \theta )\] and \[y=a\,(1-\cos \theta )\] \[\frac{dx}{d\theta }=a\,(1-\cos \theta )\] \[\frac{dy}{d\theta }=a\sin \theta \] \[\therefore \] \[\frac{dy}{dx}=\frac{dy}{d\theta }\times \frac{d\theta }{dx}=\frac{a\sin \theta }{a\,(1-\cos \theta )}\] \[=\frac{\sin \theta }{1-\cos \theta }\] \[=\frac{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}{2{{\sin }^{2}}\frac{\theta }{2}}\] [\[\sin 2\theta =2\sin \theta \cos \theta \]and \[1-{{\cos }^{2}}\theta =2{{\sin }^{2}}\theta \]] \[=\cot \frac{\theta }{2}\]
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