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]