Prove some identities of trig functions

Prove the following identities:

Proof. We use the formula for the cosine of a sum and of a difference in the following:
$\begin{align*} 2 \cos x \cos y &= \cos x \cos y + \cos x \cos y \\ &= \cos x \cos y + \sin x \sin y + \cos x \cos y - \sin x \sin y \\ &= \cos (x-y) + \cos (x+y). \qquad \blacksquare \end{align*}$
Proof. Again, we use the formula for the cosine of a sum and of a difference in the following:
$\begin{align*} 2 \sin x \sin y &= \sin x \sin y + \sin x \sin y \\ &= \cos x \cos y + \sin x \sin y - (\cos x \cos y - \sin x \sin y) \\ &= \cos (x-y) - \cos (x+y). \qquad \blacksquare \end{align*}$
Proof. Here we use the formula for the sine of a sum and difference:
$\begin{align*} 2 \sin x \cos y &= \sin x \cos y + \sin x \cos y \\ &= \sin x \cos y - \sin y \cos x + \sin x \cos y + \sin y \cos x \\ &= \sin (x-y) + \sin (x+y). \qquad \blacksquare \end{align*}$

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