Question

Find value of the sum $\frac{1}{sin{45}^{o}sin{46}^o}+\frac{1}{sin{47}^{o}sin{48}^o}+\frac{1}{sin{49}^{o}sin{50}^o}+.......+\frac{1}{sin{133}^{o}sin{134}^o}$

Answer 1

$\frac{1}{\sin 45\sin 46}+\frac{1}{\sin 46\sin 47}+....+\frac{1}{\sin 133\sin 134}$

$\Rightarrow$ Multiply and divide by $\sin 1$ we get

$=\frac{1}{\sin 1}[\frac{\sin (46-45)}{\sin 45\sin 46}+\frac{\sin (48-47)}{\sin 47\sin 48}+...+\frac{\sin (134-133)}{\sin 133\sin 134}]$

$\Rightarrow \frac{1}{\sin 1}[\cot 45-\cot 46+\cot 47-\cot 48+....+\cot 133-\cot 134]$

= we know that $\cot (\pi -\theta )=-\cot \theta$

using this we get

$\Rightarrow \frac{1}{\sin 1}[\cot 45-\cot 46+\cot 47-\cot \mathrm{48.....}+\cot 48-\cot 47+\cot 46]$

All terms cancel out except first one

$\Rightarrow \frac{\cot 45}{\sin 1}=\frac{1}{\sin 1}$.