Prove that for every pair of numbers x and h

[eqn: □sin(x+h)-(sinx+hcosx)□ LEQ {h ^{2}} over {2}]

proof) For given x, put [eqn: f(h)=sin(x+h)-sin`x-h`cos`x]

[eqn: ｜f'(h)｜=｜cos(x+h)-cos`x｜# ``````````````````````````````````=｜-2sin {2x+h} over {2} sin {h} over {2} ｜# ``````````````````````````````````=2｜sin {2x+h} over {2} ｜｜sin {h} over {2} ｜# ````````````````````````````````` LEQ 2`․`1․`｜ {h} over {2} ｜```(because`｜sin`x｜ LEQ ｜x｜`for` FORALL x)# `````````````````````````````````=｜h｜]

in other words, [eqn: -｜h｜ LEQ f'(h) LEQ ｜h｜].

Since [eqn: f(h)=f(0)+ int _{0} ^{h} {f'(t)dt`= int _{0} ^{h} {f'(t)dt}}],

[eqn: - int _{0} ^{h} {｜t｜dt LEQ f(h)= int _{0} ^{h} {f'(t)dt LEQ int _{0} ^{h} {｜t｜dt}}}].

∴[eqn: - {｜h ^{2} ｜} over {2} LEQ f(h) LEQ {｜h ^{2} ｜} over {2}],   in other words, [eqn: ｜f(h)｜ LEQ {h ^{2}} over {2}].

Therefore  for every pair of numbers x and h,  [eqn: □sin(x+h)-(sinx+hcosx)□ LEQ {h ^{2}} over {2}].

-q. e. d