Prove that xcosx< sinx for 0<x<(5π/4)

Question

-differentiate the function f(x)=sinx-xcosx

Answer 1

If we let f(x) = sinx - xcosx,

f(0) = 0
f'(x) = cos x - cosx + xsinx = xsinx

Now, f(5π/4) = -1/√2 + 5π/4 * 1/√2
= 1/√2 (5π/4 - 1)
so, f(x) is still > 0 at the end of the interval

f'(x) = 0 at x=π, so that's a maximum

so, since f is decreasing on [π,5π/4], but still positive, I'd say that

xcosx < sinx on the whole interval.