# The sum 1/(sin46sin47)+1/(sin47sin48)+1/(sin48sin49)+cdots+1/(sin133sin134)= ?

Jul 29, 2018

$\frac{1}{\sin \left(r\right) \sin \left(r + 1\right)} = \frac{1}{\sin} 1 \left[\sin \frac{1}{\sin \left(r\right) \sin \left(r + 1\right)}\right]$

$= \frac{1}{\sin} 1 \left[\sin \frac{\left(r + 1\right) - r}{\sin \left(r\right) \sin \left(r + 1\right)}\right]$

$= \frac{1}{\sin} 1 \left[\frac{\sin \left(r + 1\right) \cos \left(r\right) - \cos \left(r + 1\right) \sin \left(r\right)}{\sin \left(r\right) \sin \left(r + 1\right)}\right]$

$= \frac{1}{\sin} 1 \left[\frac{\sin \left(r + 1\right) \cos \left(r\right)}{\sin \left(r\right) \sin \left(r + 1\right)} - \frac{\cos \left(r + 1\right) \sin \left(r\right)}{\sin \left(r\right) \sin \left(r + 1\right)}\right]$

$= \frac{1}{\sin} 1 \left[\cot \left(r\right) - \cot \left(r + 1\right)\right]$

Now

$L H S = {\sum}_{r = 46}^{r = 133} \frac{1}{\sin \left(r\right) \sin \left(r + 1\right)}$

$= \frac{1}{\sin} 1 {\sum}_{r = 46}^{r = 133} \left[\cot \left(r\right) - \cot \left(r + 1\right)\right]$

$= \frac{1}{\sin} 1 \left(\cot 46 - \cot 134\right)$

$= \frac{1}{\sin} 1 \left(\cot 46 - \cot \left(180 - 46\right)\right)$

$= \frac{1}{\sin} 1 \left(\cot 46 + \cot 46\right)$

$= 2 \cot 46 \csc 1$