Let us assume the functions
f(x) and
g(x) given by
f(x)=tanx−x and
g(x)=x−sinx, for
0<x<2π
Now,
\mathrmf′(x)=sec2x−1 and
\mathrmg′(x)=1−cosx⇒f(x)>0 and
\mathrmg′(x)>0,∀x∈(0,2π)⇒f(x)>f(0) and
g(x)>g(0)∀x∈(0,2π)⇒tanx−x>0 and
x−sinx>0,∀x∈(0,2π)⇒tanx>x and
\mathrm{x} \gt \sin \mathrm{x}, \forall \mathrm{x} \in\left(0, \frac{\pi}{2}\right)\Rightarrow \quad \sin \mathrm{x}\lt \mathrm{x}\lt \tan \mathrm{x}, \forall \mathrm{x} \in \left(0,\frac{\pi}{2}\right)