An even function is symmetrical about the y-axis. In other words, for an even function \(f\), \(f(-x) = f(x)\). Cosine is one such even function, so
$$\cos(-x) = \cos(x)$$
for all real values of \(x\). This knowledge allows us to do some cool things.
Example 1: The argument of the cosine function can be anything and the outer function will remain even. Thus
$$\cos(-g(x)) = \cos(g(x))$$
for any function in the reals \(g(x)\). If we plug in a constant, we get a statement we know is true.
$$\cos \left(-\frac{\pi}{2}\right) = \cos \left(\frac{\pi}{2}\right)$$
We can also plug in a function, such as \(g(x) = x^2\).
$$\cos(-x^2) = \cos(x^2)$$
Example 2: If we have a polynomial inner function inside the cosine function with multiple terms, we must be sure to distribute the negative sign. While it is true that
$$\cos(-x^2 - x) = \cos(x^2 + x)$$
for real values of \(x\).
But, if we do not distribute the negative sign to all terms of \(g(x)\) on the left side of the equation, we instead get
$$\cos(-x^2 + x) = \cos(x^2 + x)$$
which is false.
Challenge your students to find a counterexample, a value of \(x\) for which the previous statement is not true.