Question-105414

Question Number 105414 by qwertyu last updated on 28/Jul/20

Commented by qwertyu last updated on 28/Jul/20

p l e a s e

Commented by PRITHWISH SEN 2 last updated on 28/Jul/20

\frac{1}{1 + \cot 1} + \frac{1}{1 + \cot 89} + \frac{1}{1 + \cot 5} + \frac{1}{1 + \cot 85} + \dots .

\dots + \frac{1}{1 + \cot 41} + \frac{1}{1 + \cot 49} + \frac{1}{1 + \cot 45}

= \frac{1}{1 + \cot 1} + \frac{1}{1 + \tan 1} + \frac{1}{1 + \cot 5} + \frac{1}{1 + \tan 5} + \dots .

. . + \frac{1}{1 + \cot 41} + \frac{1}{1 + \tan 41} + \frac{1}{1 + 1}

= (\frac{1}{1 + \cot 1} + \frac{\cot 1}{1 + \cot 1}) + (\frac{1}{1 + \cot 5} + \frac{\cot 5}{1 + \cot 5}) + \dots .

. . + (\frac{1}{1 + \cot 41} + \frac{\cot 41}{1 + \cot 41}) + \frac{1}{2}

= 11 + \frac{1}{2} = 11 \frac{1}{2}