Question 1107661
please help me to prove this. Thank you. 
(cos(1/2 x)+sin(1/2 x))/(cos (1/2 x)-(sin(1/2 x))=secx+tanx
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Half-angle formulas are usually messier than double angle formuals.
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Start with this:
{{{(cos(x)+sin(x))/(cos(x)-sin(x)) = sec(2x) + tan(2x) = (1 + sin(2x))/cos(2x)}}}
If that's confusing, sub y for x/2:
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{{{(cos(y)+sin(y))/(cos(y)-sin(y)) = (1 + sin(2y))/cos(2y)}}}
cos(2y) = cos^2(y) - sin^2(y) = (cos(y)-sin(y))*(cos(y)+sin(y))
Multiply by (cos(y)-sin(y))
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{{{cos(y)+sin(y) = (1 + sin(2y))/(cos(y)+sin(y))}}}
{{{cos^2(y)+2sin(y)*cos(y)+sin^2(y) = 1 + sin(2y) = 1 + 2sin(y)*cos(y)}}}
QED