数学三角函数恒等变化

题目

数学三角函数恒等变化
怎么用只含有半角的正切的式子分别表示角的正弦、余弦和正切,

答案

sinx=2sin(x/2)cos(x/2)=2tan(x/2)cos^2(x /2);
cosx=cos^2(x /2)-sin^2 (x /2)=(1-tan^2(x /2))cos^2 (x /2);
tanx=2tan(x /2)/(1-tan^2 (x /2));
cos^2(x /2)+sin^2 (x /2)=1;
1+tan^2 (x /2))=1/cos^2 (x /2);
故cos^2(x /2)=1/(tan^2(x /2)+1),代入
得sinx=2tan(x/2)cos^2(x /2)=2tan(x/2)/(tan^2(x /2)+1)
cosx=(1-tan^2(x /2))cos^2 (x /2)=(1-tan^2(x /2))/(tan^2(x /2)+1)
cotx=1/tanx=(1-tan^2 (x /2))/(2tan(x /2))