(1)
f(x)
= x - sinx
f(0)
= 0
f¢(x)
= 1-cos x ³ 0 .
Hence f(x) ³ 0 for all x ³ 0 that is x ³ sinx for x ³ 0.

(2)
f(x)
= cosx - (1 - x2
2
 )
f(0)
= 0
f¢(x)
= -sinx + x ³ 0     by (1)
Hence
cosx ³ 1 - x2
2

for x ³ 0.

(3)
f(x)
= (x - x3
3!
 ) - sinx
f(0)
= 0
f¢(x)
= 1 - x2
2
 - cosx £ 0     by (2)
Hence
sinx ³ (x - x3
3!
 )

for x ³ 0.

(4)
f(x)
= cosx - (1 - x2
2!
 + x4
4!
 )
f(0)
=0
f¢(x)
= - sinx + x - x3
3!
 £ 0     by (3)
Hence
cosx £ (1 - x2
2!
 + x4
4!
 )

for x ³ 0.

(5) This process can be repeated indefinitely to give the Maclaurin series, valid for all x (in radians)
cosx = 1 - x2
2!
 + x4
4!
 - x6
6!
 + ...

sinx = x - x3
3!
 + x5
5!
 - x7
7!
 + ...