The sum $\frac{1}{sin 46 sin 47} + \frac{1}{sin 47 sin 48} + \frac{1}{sin 48 sin 49} + \dots + \frac{1}{sin 133 sin 134} =$ $1/(sin46sin47)+1/(sin47sin48)+1/(sin48sin49)+cdots+1/(sin133sin134)=$ ?

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The sum $\frac{1}{sin 46 sin 47} + \frac{1}{sin 47 sin 48} + \frac{1}{sin 48 sin 49} + \dots + \frac{1}{sin 133 sin 134} =$ $1/(sin46sin47)+1/(sin47sin48)+1/(sin48sin49)+cdots+1/(sin133sin134)=$ ?

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Answer 1 · 2018-07-29T17:23:50

$\frac{1}{sin (r) sin (r + 1)} = \frac{1}{sin 1} [\frac{sin 1}{sin (r) sin (r + 1)}]$ $1/(sin(r)sin(r+1))=1/sin1[sin1/(sin(r)sin(r+1))]$

$= \frac{1}{sin 1} [\frac{sin ((r + 1) - r)}{sin (r) sin (r + 1)}]$ $=1/sin1[ sin((r+1)-r)/(sin(r)sin(r+1))]$

$= \frac{1}{sin 1} [\frac{sin (r + 1) cos (r) - cos (r + 1) sin (r)}{sin (r) sin (r + 1)}]$ $=1/sin1[ (sin(r+1)cos(r)-cos(r+1)sin(r))/(sin(r)sin(r+1))]$

$= \frac{1}{sin 1} [\frac{sin (r + 1) cos (r)}{sin (r) sin (r + 1)} - \frac{cos (r + 1) sin (r)}{sin (r) sin (r + 1)}]$ $=1/sin1[ (sin(r+1)cos(r))/(sin(r)sin(r+1))-(cos(r+1)sin(r))/(sin(r)sin(r+1))]$

$= \frac{1}{sin 1} [cot (r) - cot (r + 1)]$ $=1/sin1[ cot(r)-cot(r+1)]$

Now

$L H S = r = 133 \sum r = 46 \frac{1}{sin (r) sin (r + 1)}$ $LHS=sum_(r=46)^(r=133)1/(sin(r)sin(r+1))$

$= \frac{1}{sin 1} r = 133 \sum r = 46 [cot (r) - cot (r + 1)]$ $=1/sin1sum_(r=46)^(r=133)[ cot(r)-cot(r+1)]$

$= \frac{1}{sin 1} (cot 46 - cot 134)$ $=1/sin1(cot46-cot134)$

$= \frac{1}{sin 1} (cot 46 - cot (180 - 46))$ $=1/sin1(cot46-cot(180-46))$

$= \frac{1}{sin 1} (cot 46 + cot 46)$ $=1/sin1(cot46+cot46)$

$= 2 cot 46 csc 1$ $=2cot46csc1$

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The sum $\frac{1}{sin 46 sin 47} + \frac{1}{sin 47 sin 48} + \frac{1}{sin 48 sin 49} + \dots + \frac{1}{sin 133 sin 134} =$ $1/(sin46sin47)+1/(sin47sin48)+1/(sin48sin49)+cdots+1/(sin133sin134)=$ ?

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

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The sum $\frac{1}{sin 46 sin 47} + \frac{1}{sin 47 sin 48} + \frac{1}{sin 48 sin 49} + \dots + \frac{1}{sin 133 sin 134} =$ $1/(sin46sin47)+1/(sin47sin48)+1/(sin48sin49)+cdots+1/(sin133sin134)=$ ?