mean value theorem sin(b) - sin(a)

Question

It's too much hassle to post it here as latex, to so here's the screenshot.

I don't understand why |cos(c)| = 1

Why 1? Why not $\frac {\sqrt{3}}{2}$? Why absolute value assumes the max value a function can take? Shouldn't it be like:

$\cos(c) > 0$

and $-\cos(c) < 0$

?

Answer 1

Where do you see $|\cos(c)|=1$? I see $|\cos(c)|\leq1$ being used, which is a well-known property of the $\cos$ function.