Proving of Inequalities by Contradiction

The essence of this method is in the following: suppose we need to prove inequality f(x,y,z)>g(x,y,z). We assume contrary proposition, i.e. that for some x,y and z f(x,y,z)<=g(x,y,z).

Using properties of inequalities we perform transformations of inequality. If as result, we obtain incorrect inequality than this means that our assumption that f(x,y,z)<=g(x,y,z) was wrong, therefore, f(x,y,z)>g(x,y,z).

Example. Prove that cos(alpha+beta)cos(alpha-beta)<=cos^2(alpha).

Assume contrary proposition, i.e. assume that exist some alpha and beta such that cos(alpha+beta)cos(alpha-beta)>cos^2(alpha).

Using formulas cos(alpha+beta)cos(alpha-beta)=(cos(2beta)+cos(2alpha))/2 and cos^2(alpha)=(1+cos(2alpha))/2, we obtain that (cos(2beta)+cos(2alpha))/2>(1+cos(2alpha))/2. From this we have that cos(2beta)>1.

Since for any beta cos(2beta)<=1 then we obtained contradiction. Thus, our asumption is incorrect, therefore, cos(alpha+beta)cos(alpha-beta)<=cos^2(alpha).