已知sin(2a+x)=2sinx.求证：tan(a+x)=3tana

题目

已知sin(2a+x)=2sinx.求证：tan(a+x)=3tana

答案

因为
sin(2a+x) = sin(a+x+a) = sin(a+x)cos(a)+cos(a+x)sin(a)
2sin(x) = 2sin(a+x-a) = 2sin(a+x)cos(a)-2cos(a+x)sin(a)
sin(2a+x)=2sin(x)
所以整理可得
sin(a+x)cos(a) = 3cos(a+x)sin(a)
即
sin(a+x)/cos(a+x) = 3 sin(a)/cos(a)
可证
tan(a+x) = 3 tan(a)