How to prove that $\cos x< \cos (\sin x)$ using the mean value theorem?

Question

Using the mean value theorem, prove that, for $0<x<\pi/2$, $\cos x<\cos(\sin x)$.

I was trying to use the mean value theorem but I got lost. I am a newbie to this please explain this as you're explaining it to your worst student!

Answer 1

HINT

Note that

• $x>\sin x$
• $\cos x$ is a strictly decreasing function on the interval

Answer 2

You can show that $\cos(x)$ is decreasing on that and then use the fact that $x > \sin(x)$ (you may have to show this as well, but that is not a challenge).