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The smallest integer $n$ such that $\frac{1}{\sin 45^{\circ} \sin 46^{\circ}}+\frac{1}{\sin 47^{\circ} \sin 48^{\circ}}+\ldots$ $+\frac{1}{\sin 133^{\circ} \sin 134^{\circ}}=\frac{1}{\sin \left(n^0\right)}$ is
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$1$
$\begin{aligned} \frac{1}{\sin 45^{\circ} \cdot \sin 46^{\circ}}+\frac{1}{\sin 47^{\circ} \cdot \sin 48^{\circ}}+\ldots+ \\ \frac{1}{\sin 133^{\circ} \cdot \sin 134^{\circ}}\end{aligned}$
$\begin{array}{r}=\frac{1}{\sin 1^{\circ}}\left[\frac{\sin \left(46^{\circ}-45^{\circ}\right)}{\sin 45^{\circ} \cdot \sin 46^{\circ}}+\frac{\sin \left(48^{\circ}-47^{\circ}\right)}{\sin 47^{\circ} \cdot \sin 48^{\circ}}+\ldots+\right. \\ \left.\frac{\sin \left(134^{\circ}-133^{\circ}\right)}{\sin 133^{\circ} \cdot \sin 134^{\circ}}\right]\end{array}$
$\begin{array}{r}=\frac{1}{\sin 1^{\circ}}\left[\left(\cot 45^{\circ}-\cot 46^{\circ}\right)+\left(\cot 47^{\circ}-\cot 48^{\circ}\right)\right. \\ \left.+\ldots+\left(\cot 133^{\circ}-\cot 134^{\circ}\right)\right]\end{array}$
$=\frac{1}{\sin 1^{\circ}} \times \cot 45^{\circ}=\frac{1}{\sin 1^{\circ}}$
According to question,
$\begin{array}{rlrl} & & \frac{1}{\sin \left(n^{\circ}\right)} & =\frac{1}{\sin 1^{\circ}} \\ \therefore \quad n & n & =1\end{array}$
$\begin{array}{r}=\frac{1}{\sin 1^{\circ}}\left[\frac{\sin \left(46^{\circ}-45^{\circ}\right)}{\sin 45^{\circ} \cdot \sin 46^{\circ}}+\frac{\sin \left(48^{\circ}-47^{\circ}\right)}{\sin 47^{\circ} \cdot \sin 48^{\circ}}+\ldots+\right. \\ \left.\frac{\sin \left(134^{\circ}-133^{\circ}\right)}{\sin 133^{\circ} \cdot \sin 134^{\circ}}\right]\end{array}$
$\begin{array}{r}=\frac{1}{\sin 1^{\circ}}\left[\left(\cot 45^{\circ}-\cot 46^{\circ}\right)+\left(\cot 47^{\circ}-\cot 48^{\circ}\right)\right. \\ \left.+\ldots+\left(\cot 133^{\circ}-\cot 134^{\circ}\right)\right]\end{array}$
$=\frac{1}{\sin 1^{\circ}} \times \cot 45^{\circ}=\frac{1}{\sin 1^{\circ}}$
According to question,
$\begin{array}{rlrl} & & \frac{1}{\sin \left(n^{\circ}\right)} & =\frac{1}{\sin 1^{\circ}} \\ \therefore \quad n & n & =1\end{array}$
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