0

I found this question here (I recommend that you read the question and the highest-voted answer there) How to solve for $x$ in $x(x^3+\sin x \cos x)-\sin^2 x =0$? and the math below is an answer. I guess I have 2 questions in 1 here.

First, how did the guy suddenly realize that $$x^4+x\sin(x)\cos(x)−\sin^2(x)>x^2\sin^2(x)+\sin^2(x\cos(x)−\sin^2(x)$$ I don't see why this statement is necessarily true.

My second question: How did he construct, out of nowhere, a random inequality like this which ended up being useful? Thanks.

Community
  • 1
Ovi
  • 24,817
  • 1
    As far as verifying the inequality, if you check term by term you should be able to see that he has substituted $\sin x$ for $x$ in ways that decrease the expression. That gave him $\sin^2 x $ that he could pull out. It looks clever to me, too. – Ross Millikan Apr 29 '13 at 00:56

1 Answers1

1

In this solution, he restricted to values of $x$ where $x>\sin x$ and $\cos x>0$.

vadim123
  • 83,937