Prove or disprove some trig identities

For each of the following, either prove the identity or provide a counterexample to show the identity is not always true.

For all real $x \neq 0$ , $\sin 2x \neq 2 \sin x$ .
For all real $x$ , there exists a real $y$ such that $\cos (x+y) = \cos x + \cos y$ .
For every real $y$ , there exists a real $x$ such that $\sin (x+y) = \sin x + \sin y$ .
There exists a real $y \neq 0$ such that $\int_0^y \sin x \, dx = \sin y$ .

False.
Counterexample: Let $x = \pi$ , then $\sin 2x = \sin 2 \pi = 0$ , and $2 \sin x = 2 \sin \pi =0$ . Hence, we have $x \neq 0$ , but $\sin 2x = 2 \sin x$ ; thus, the statement is false.
False.
Counterexample: Let $x = 0$ , then $\cos (x+y) = \cos y$ , but $\cos x + \cos y = 1 + \cos y$ .
True.
Proof. Let $x = 0$ , then $\sin (x+y) = \sin y$ for all $y$ , and $\sin x + \sin y = \sin y$ for all $y. \qquad \blacksquare$
True.
Proof. Let $y = \frac{\pi}{2}$ , then

$\int_0^{\pi}{2} \sin x \, dx = 1 - \cos \frac{\pi}{2} = 1 \qquad \text{and} \qquad \sin \frac{\pi}{2} = 1. \qquad \blacksquare$

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