Hello all

I am trying to prove that

\[sin(x)-cos(x)\geq 1\]

For each x in the interval \[[\frac{\pi }{2},\pi ]\]

I tried doing it by contradiction, what I did was:

Assume

\[sin(x)-cos(x)< 1\]

Then I used the power of 2 on each side of the inequality and got:

\[sin^{2}(x)-2sin(x)cos(x)+cos^{2}(x)<1\]

which led me to

\[0<2sin(x)cos(x)\]

which is a contradiction since

\[cos(\frac{\pi }{2})=0\]

I am not sure that what I did is correct or complete. Can you please check my proof and give me your opinion on the matter?

Thank you in advance.