证明1+sinx/cosx=tan(π/4+x/2)
问题描述：
证明1+sinx/cosx=tan(π/4+x/2)
答
tan(π/4+x/2)
=(tanπ/4+tanx/2)/(1-tanπ/4*tanx/2)
=(1+tanx/2)/(1-tanx/2)
=(cosx/2+sinx/2)/(cosx/2-sinx/2)
=(cosx/2+sinx/2)(cosx/2+sinx/2)/(cosx/2+sinx/2)/(cosx/2-sinx/2)
=(1+2sinx/2cosx/2)/(cos^2 x/2-sin^2 x/2)
=(1+sinx)/cosx
答
令x/2=a
则左边=(1+sin2a)/cos2a
=(sin²a+cos²a+2sinacosa)/(cos²a-sin²a)
=(cosa+sina)²/[(cosa+sina)(cosa-sina)]
=(cosa+sina)/(cosa-sina)
上下除以cosa
=(1+tana)/(1-tana)
=(tanπ/4-tana)/(1-tanπ/4tana)
=tan(π/4-a)=右边
命题得证