Question

Proove that (cos⁡(x)-cos^2⁡(x)+1 ≥ 1) can also be written as (1 ≥ cos⁡(x)+1 ≥ 1)

Answer 1

Answer:

1

I am trying to prove that sin(x)−cos(x)≥1 for every x in the interval [

π

2

,π].

I started by assuming that it is false, i.e. there exists an x for which sin(x)−cos(x)<1. In the next step I got stuck, since I wanted to take the square of each side of the inequality, so I can get [sin(x)−cos(x)]2<1, but this is not true, since a<b doesn't imply a2<b2. Can you assist me to prove this statement by contradiction? Is there also a way to prove it without contradiction? Thank you.

trigonometry proof-writing