Question

Prove that (1-cos a +sin a)/(1+cos a +sin a) = tan a/2
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Answer 1

cosa=cos²(a/2)-sin²(a/2)

Sina=2sin(a/2)cos(a/2)

1-cos²(a/2)=sin²(a/2)

1-sin²(a/2)=cos²(a/2)

by substituting

LHS=(1-(cos²(a/2)-sin²(a/2))+2sin(a/2)cos(a/2))/(1+cos²(a/2)-sin²(a/2)

=(2sin²(a/2)+2sin(a/2)cos(a/2))/(2cos²(a/2)+2sin(a/2)cos(a/2))

sin(a/2)(sin(a/2)+cos(a/2))/(cos(a/2)(sin(a/2)+cos(a/2))

=tan(a/2)