以下推导过程中，有误的是（   ）

A：$\begin{array}{c}\sin \left(\alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\ \sin \left(\alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta \end{array}\}\Rightarrow \sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)=2\sin \alpha \cos \beta \Rightarrow \sin \alpha \cos \beta =\frac{1}{2}\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)$

B：$\cos \alpha ={\cos }^2\frac{\alpha }{2}-{\sin }^2\frac{\alpha }{2}=\frac{{\cos }^2\frac{\alpha }{2}-{\sin }^2\frac{\alpha }{2}}{{\cos }^2\frac{\alpha }{2}+{\sin }^2\frac{\alpha }{2}}=\frac{1-{\tan }^2\frac{\alpha }{2}}{1+{\tan }^2\frac{\alpha }{2}}$

C：$\begin{array}{c}\cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta \\ \sin \left(\alpha -\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta \end{array}\}\Rightarrow \cos \left(\alpha +\beta \right)-\cos \left(\alpha -\beta \right)=2\sin \alpha \sin \beta \Rightarrow \sin \alpha \cos \beta =\frac{1}{2}\cos \left(\alpha +\beta \right)-\cos \left(\alpha -\beta \right)$

D：$\sin \alpha =2\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}=2\tan \frac{\alpha }{2}{\cos }^2\frac{\alpha }{2}=\frac{2\tan \frac{\alpha }{2}{\cos }^2\frac{\alpha }{2}}{{\sin }^2\frac{\alpha }{2}+{\cos }^2\frac{\alpha }{2}}=\frac{2\tan \frac{\alpha }{2}}{{\tan }^2\frac{\alpha }{2}+1}$