$\cos\frac{\pi}{7}$

Adam claims that $\cos\frac{\pi}{7}$ is equal to each of the following four expressions: $$\cos⁡\left(-\frac{\pi}{7}\right),\quad \cos\frac{13\pi}{7},\quad\cos\frac{⁡8\pi}{7},\quad\sin⁡\frac{9\pi}{14}$$

He says that:

(1) Cosine is an even function, so $$\cos⁡\frac{\pi}{7}=\cos⁡\left(-\frac{\pi}{7}\right).$$

(2) Cosine is an even function and periodic with period $2\pi$, so $$\cos\frac{\pi}{7}=\cos\left(-\frac{\pi}{7}\right)=\cos⁡\left(-\frac{\pi}{7}+2\pi\right)=\cos\frac{13\pi}{7}.$$

(3) It holds that $\cos⁡x=\cos⁡(x+\pi)$ for every real number $x$, so $$\cos\frac{⁡\pi}{7}=\cos\left(\frac{\pi}{7}+\pi\right)=\cos⁡\frac{8\pi}{7}.$$

(4) It holds that $\cos⁡x=\sin\left(x+\frac{\pi}{2}\right)$ for every real number $x$, so $$\cos⁡\frac{\pi}{7}=\sin⁡\left(\frac{\pi}{7}+\frac{\pi}{2}\right)=\sin⁡\frac{9\pi}{14}.$$

Adam's classmates commented on his statements:

• John: “Adam is right.”
• Elisabeth: “Adam's statements (1) and (2) are false. Cosine is an odd function.”
• Paul: “Adam's statement (2) is false. Cosine is a periodic function with period $\pi$.”
• Mary: “Adam's statement (3) is false. It holds that $\cos⁡x=-\cos(x+\pi)$ for every real number $x$.”
• Lucy: “Adam's statement (4) is false. It holds that $\cos⁡x=\sin\left(x-\frac{\pi}{2}\right)$ for every real number $x$.”

Identify which of Adam's classmates is correct.