The smallest integer n such that frac{1}{sin | vedclass

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Question

(A) Let the given sum be $S$. The series is $S = \sum_{k=0}^{44} \frac{1}{\sin(45^{\circ}+2k) \sin(46^{\circ}+2k)}$.Multiplying and dividing by $\sin

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(A) Let the given sum be $S$. The series is $S = \sum_{k=0}^{44} \frac{1}{\sin(45^{\circ}+2k) \sin(46^{\circ}+2k)}$.Multiplying and dividing by $\sin 1^{\circ}$,we get:$S = \frac{1}{\sin 1^{\circ}} \sum_{k=0}^{44} \frac{\sin((46^{\circ}+2k)-(45^{\circ}+2k))}{\sin(45^{\circ}+2k) \sin(46^{\circ}+2k)}$$S = \frac{1}{\sin 1^{\circ}} \sum_{k=0}^{44} (\cot(45^{\circ}+2k) - \cot(46^{\circ}+2k))$Expanding the sum:$S = \frac{1}{\sin 1^{\circ}} [(\cot 45^{\circ} - \cot 46^{\circ}) + (\cot 47^{\circ} - \cot 48^{\circ}) + \ldots + (\cot 133^{\circ} - \cot 134^{\circ})]$Since $\cot(180^{\circ}-	heta) = -\cot 	heta$,we have $\cot 133^{\circ} = -\cot 47^{\circ}$ and $\cot 134^{\circ} = -\cot 46^{\circ}$.Substituting these values,the terms cancel out:$S = \frac{1}{\sin 1^{\circ}} [\cot 45^{\circ} - \cot 46^{\circ} + \cot 47^{\circ} - \cot 48^{\circ} + \ldots - \cot 47^{\circ} + \cot 46^{\circ}]$$S = \frac{1}{\sin 1^{\circ}} (\cot 45^{\circ}) = \frac{1}{\sin 1^{\circ}}$.Given $\frac{1}{\sin(n^{\circ})} = \frac{1}{\sin 1^{\circ}}$,we get $n = 1$.

The smallest integer $n$ such that $\frac{1}{\sin 45^{\circ} \sin 46^{\circ}}+\frac{1}{\sin 47^{\circ} \sin 48^{\circ}}+\ldots+\frac{1}{\sin 133^{\circ} \sin 134^{\circ}}=\frac{1}{\sin \left(n^{\circ}\right)}$ is

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