# Trigonometric equality

I would like to know, how do you simplify this: $$\cos x\sin(x+y) + \sin x\cos(x+y)$$ to this: $$\sin(2x+y).$$

Wolfram alpha says so, but how does human being do so? :)

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I hope you are aware of the $\sin(A+B)$ formula which is $$\sin(A+B)=\sin{A}\cos{B}+ \cos{A}\sin{B}$$.

For a complete list of Trigonometric identities please see:

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A human being uses the addition formula for the sine $$\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$$ And applies it with $\alpha=x+y$ and $\beta=x$.
We know that $\sin(x+y) = \sin x \cos y + \cos x \sin y$ and $\cos(x+y) = \cos x \cos y - \sin x \sin y$.