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1-cosx/2 / 1+cosx/2  

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Supposing that you need to simplify the expression `((1-cos x)/2))/((1 + cos x)/2))` , you should use the half angle formulas such that:

`sin(x/2) = sqrt((1-cos x)/2) => sin^2(x/2) = (1-cos x)/2`

`cos(x/2) = sqrt((1+cos x)/2) => cos^2(x/2) = (1+cos x)/2`

Hence, substituting `sin^2(x/2)`  for `(1-cos x)/2`  and `(1+cos x)/2`  for `cos^2(x/2)`  yields:

`((1-cos x)/2)/((1+cos x)/2) = (sin^2(x/2))/(cos^2(x/2))`

You need to remember that sin alpha/cos alpha = tan alpha, hence, reasoning by analogy yields:

`(sin^2(x/2))/(cos^2(x/2)) = (sin(x/2))/(cos(x/2))*(sin(x/2))/(cos(x/2)) = tan(x/2)*tan(x/2) = tan^2 (x/2)`

Hence, simplifying the given expression yields `((1-cos x)/2)/((1+cos x)/2) = tan^2 (x/2).`

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