设tan(θ/2)=t,求证：sinθ=2t/(1+t^2),cosθ=(1-t^2)/(1+t^2),tanθ=2t/(11t^2)
问题描述：
设tan(θ/2)=t,求证：sinθ=2t/(1+t^2),cosθ=(1-t^2)/(1+t^2),tanθ=2t/(11t^2)
设tan(θ/2)=t,求证：sinθ=2t/(1+t^2),cosθ=(1-t^2)/(1+t^2),tanθ=2t/(1-t^2)
答
sinθ=2sinθ/2*cosθ/2
=2tanθ/2*cosθ/2*cosθ/2
=2tanθ/2(cosθ/2)^2
=2t*(cosθ/2)^2
而（cosθ/2)^2
=1/（secθ/2)^2
=1/[1+（tanθ/2)^2]
=1/1+t^2
所以sinθ=2t/(1+t^2)
同理
cosθ=2(cosθ/2)^2-1
=2/1+t^2-1
=(1-t^2)/(1+t^2)
tanθ=sinθ/cosθ=2t/(1-t^2)