Rút gọn A=[1+cos(x/2)-sin(x/2)]/[1-cos(x/2)-sin(x/2)]

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1. Đáp án:

   $A =  – \cot \left( {\dfrac{x}{4}} \right)$

   Giải thích các bước giải:

   $\begin{array}{l}
   A = \dfrac{{1 + \cos \left( {\dfrac{x}{2}} \right) – \sin \left( {\dfrac{x}{2}} \right)}}{{1 – \cos \left( {\dfrac{x}{2}} \right) – \sin \left( {\dfrac{x}{2}} \right)}}\\
    = \dfrac{{2{{\cos }^2}\left( {\dfrac{x}{4}} \right) – 2\sin \left( {\dfrac{x}{4}} \right)\cos \left( {\dfrac{x}{4}} \right)}}{{2{{\sin }^2}\left( {\dfrac{x}{4}} \right) – 2\sin \left( {\dfrac{x}{4}} \right)\cos \left( {\dfrac{x}{4}} \right)}}\\
    = \dfrac{{2\cos \left( {\dfrac{x}{4}} \right)\left( {\cos \left( {\dfrac{x}{4}} \right) – \sin \left( {\dfrac{x}{4}} \right)} \right)}}{{2\sin \left( {\dfrac{x}{4}} \right)\left( {\sin \left( {\dfrac{x}{4}} \right) – \cos \left( {\dfrac{x}{4}} \right)} \right)}}\\
    =  – \dfrac{{\cos \left( {\dfrac{x}{4}} \right)}}{{\sin \left( {\dfrac{x}{4}} \right)}}\\
    =  – \cot \left( {\dfrac{x}{4}} \right)
   \end{array}$

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