Question

demonstrati sinx-cosx <=radical din 2, oricare x apartine R

Answer 1

E = sinx - cosx

E² = sin²x + cos²x - 2sinxcosx

E² = 1-sin(2x)

-1 ≤ sin(2x) ≤ 1|•(-1)

1 ≥ -sin(2x) ≥ -1

2 ≥ 1-sin(2x) ≥ 0

=> 2 ≥ E² ≥ 0 => 0 ≤ E² ≤ 2 =>

=> E² ≤ 2 => -√2 ≤ E ≤ √2

=> sinx - cosx ≤ √2