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

Commented by qwertyu last updated on 28/Jul/20

please

$$\boldsymbol{{please}} \\ $$

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

(1/(1+cot 1))+(1/(1+cot 89)) +(1/(1+cot 5)) +(1/(1+cot 85)) +.... ...+(1/(1+cot 41))+(1/(1+cot 49)) +(1/(1+cot 45)) =(1/(1+cot 1))+(1/(1+tan 1))+(1/(1+cot 5))+(1/(1+tan 5))+.... ..+(1/(1+cot 41))+(1/(1+tan 41))+(1/(1+1)) =((1/(1+cot 1))+((cot 1)/(1+cot 1)))+((1/(1+cot 5))+((cot 5)/(1+cot 5)))+.... ..+((1/(1+cot 41))+((cot 41)/(1+cot 41)))+(1/2) =11+(1/2) = 11(1/2)

$$\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{89}}\:+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{5}}\:+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{85}}\:+.... \\ $$$$...+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{41}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{49}}\:+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{45}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{tan}\:\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{tan}\:\mathrm{5}}+.... \\ $$$$..+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{41}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{tan}\:\mathrm{41}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{1}} \\ $$$$=\left(\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{1}}+\frac{\mathrm{cot}\:\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{1}}\right)+\left(\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{5}}+\frac{\mathrm{cot}\:\mathrm{5}}{\mathrm{1}+\mathrm{cot}\:\mathrm{5}}\right)+.... \\ $$$$..+\left(\frac{\mathrm{1}}{\mathrm{1}+\mathrm{cot}\:\mathrm{41}}+\frac{\mathrm{cot}\:\mathrm{41}}{\mathrm{1}+\mathrm{cot}\:\mathrm{41}}\right)+\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$=\mathrm{11}+\frac{\mathrm{1}}{\mathrm{2}}\:=\:\mathrm{11}\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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