Question

For all real $\theta$, prove $\cos (\cos \theta$) > 0

Answer 1

For all real $\theta$, we have $-1\le cos\theta \le 1$
Let $cos\theta =A$ so that $-1\le A\le 1$ Extend the interval.
Since $-\frac{\pi }{2}<-1and1<\frac{\pi }{2}$, it follows that
$-\frac{\pi }{2}<A<\frac{\pi }{2}$
Hence $cosA>0$, that is, $cos\left(cos\theta \right)>0$