>>133

1)　tan(x) = sin(x)/cos(x)
　　　= {x -(1/3!)x^3 +(1/5!)x^5 - ･･･}/{1 -(1/2!)x^2 +(1/4!)x^4 -(1/6!)x^6 + ･･･}
　　　= x + (1/3)x^3 + (2/15)x^5 + O(x^7)

2)　sin(x) = x - (1/3!)x^3 + (1/5!)x^5 - ････,
　x･sin(x) = x^2 - (1/3!)x^4 + (1/5!)x^6 - ････,
　e^(x･sin(x)) = 1 + x･sin(x) + (1/2!){x･sin(x)}^2 + (1/3!){x･sin(x)}^3 + ････
　　　　　　　= 1 + x＾2 +(1/3)x^4 + (1/120)x^6 + O(x^8),

3)　x/(e^x -1) = x/{Σ[k=1,∞] (1/k!)x^k }
　　　　　= 1/{1 + (1/2)x + (1/6)x^2 + (1/24)x^3 + (1/120)x^4 + ････ }
　　　　　= 1 -(1/2)x +(1/12)x^2 -(1/720)x^4 + O(x^6)

(別解)
　x/(e^x -1) + (x/2) は偶関数。
　x/(e^x -1)= (x/2)(e^x +1)/(e^x -1) - (x/2)
　　　　　　= (x/2){e^(x/2) + e^(-x/2)}/{e^(x/2) - e^(-x/2)} - (x/2)
　　　　　　= (x/2)coth(x/2) - (x/2)
　　　　　　= 1 -(1/2)x +(1/12)x^2 -(1/720)x^4 + O(x^6)