Real Cosine Function is Bounded

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Let $x \in \R$.


$\size {\cos x} \le 1$


From the algebraic definition of the real cosine function:

$\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}$

it follows that $\cos x$ is a real function.

Similarly, $\sin x$ is also a real function.

Thus it follows that:

\(\ds \cos^2 x\) \(\le\) \(\ds \cos^2 x + \sin^2 x\) as $\sin^2 x \ge 0$ by Square of Real Number is Non-Negative
\(\ds \) \(=\) \(\ds 1\) Sum of Squares of Sine and Cosine

From Ordering of Squares in Reals and the definition of absolute value, we have that:

$x^2 \le 1 \iff \size x \le 1$

The result follows.


Also see