Cosine

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The cosine function, $\cos \colon \mathbb{R} \rightarrow \mathbb{R}$ is the unique solution of the second order initial value problem $$y=-y;y(0)=1,y'(0)=0.$$

Properties

Proposition: $$\cos(x) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k x^{2k}}{(2k)!}$$

Proof:

Proposition: $$\cos(x) = \prod_{k=1}^{\infty} \left( 1 - \dfrac{4x^2}{\pi^2 (2k-1)^2} \right)$$

Proof: