91Ó°ÊÓ

The cosine function, denoted by cos(x), has a range of values it can output between -1 and 1, inclusive. That is, for any real number x, we have \[-1 \leq \cos x \leq 1.\]

Notice that \(\pi\) is a number greater than 2 but less than 4 (approximately 3.14) and never changes, as it's a fundamental constant. The power function \(\pi^z\) has the property that it increases as z increases when the base is greater than 1. Thus, if z is smaller, the entire expression \(\pi^z\) is smaller.

Using the information from steps 1 and 2, we can now determine the possible values of \(\pi^{\cos x}\). Since the cosine function has values between -1 and 1, we can substitute the smallest and the largest possible values for \(\cos x\) to find the smallest and largest possible values of \(\pi^{\cos x}\). For the smallest possible value of \(\cos x\), we use -1 (from the range of cosine): \[\pi^{-1} = \frac{1}{\pi},\] which is less than 1 since \(\pi > 1\). For the largest possible value of \(\cos x\), we use 1 (again, from the range of cosine): \[\pi^1 = \pi,\] which is less than 4 since \(\pi < 4\). Thus, the smallest and largest possible values of \(\pi^{\cos x}\) are between \(\frac{1}{\pi}\) and \(\pi\), which means that for any real number \(x\), the expression \(\pi^{\cos x}\) will always be less than 4.

In conclusion, we have shown that for any real number \(x\), the expression \(\pi^{\cos x}\) will always be less than 4, so the inequality \(\pi^{\cos x} < 4\) is true for every real number \(x\).