Упростите это выражение пожалуйста)

[latex] frac{1-2sin frac{x}{2}-cosx }{1+2sin frac{x}{2}-cosx }= frac{1-2sin frac{x}{2}-cos(2* frac{x}{2} ) }{1+2sin frac{x}{2}-cos(2* frac{x}{2} ) }= \ \ = frac{1-2sin frac{x}{2}-(cos^2 frac{x}{2}-sin^2 frac{x}{2} ) }{1+2sin frac{x}{2}-(cos^2 frac{x}{2}-sin^2 frac{x}{2} ) }= \ \ = frac{1-2sin frac{x}{2}-(1-sin^2 frac{x}{2}-sin^2 frac{x}{2} ) }{1+2sin frac{x}{2}-(1-sin^2 frac{x}{2}-sin^2 frac{x}{2} ) }= \ \ [/latex][latex]= frac{1-2sin frac{x}{2}-1+2sin^2 frac{x}{2} }{1+2sin frac{x}{2}-1+2sin^2 frac{x}{2} }= frac{-2sin frac{x}{2}+2sin^2 frac{x}{2} }{2sin frac{x}{2}+2sin^2 frac{x}{2} }= \ \ = frac{2sin frac{x}{2}(sin frac{x}{2}-1 ) }{2sin frac{x}{2}(sin frac{x}{2}+1 ) }= frac{sin frac{x}{2}-1 }{sin frac{x}{2}+1 } [/latex]