<s>[INST] How to verify $\frac{\sin x\pm\sin y}{\cos x+\cos y}=\tan\frac{x\pm y}{2}$ Can anyone help me verify the following identity?
$$
\frac{\sin x\pm\sin y}{\cos x+\cos y}=\tan\frac{x\pm y}{2}
$$
I could really use some guidance. 
 [/INST] $$\sin x+\sin y=2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$$
$$\sin x -\sin y=2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$$
$$\cos x+\cos y=2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$$
$$\frac{\sin x+\sin y}{\cos x+\cos y}=\frac{2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})}{2\cos(\frac{x+y}{2})\cos(\frac{x-y}{2})}=\frac{\sin(\frac{x+y}{2})}{\cos(\frac{x+y}{2})}=\tan\left(\frac{x+y}{2}\right)$$
and the same method for the other!
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