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Question Number 71984 by Henri Boucatchou last updated on 23/Oct/19
x, y ∈ I ⊂ R prove that ∣cosx − siny∣ ≤ ∣x − y∣
x,y∈I⊂Rprovethat∣cosx−siny∣⩽∣x−y∣
Answered by mind is power last updated on 23/Oct/19
x=y=0 mistack ⇒1≤0 i think is ∣sin(x)−sin(y)∣≤∣x−y∣ letf(x,y)= ∫_x ^y cos(t)dt f(x,y)=[sin(t)]_x ^y =sin(y)−sin(x) ∣f(x,y)∣=∣∫_x ^y cos(t)dt∣≤∣∫_x ^y ∣cos(t)dt∣∣≤∣∫_x ^y 1dt∣=∣y−x∣ ⇒∣sin(y)−sin(x)∣≤∣y−x∣
x=y=0mistack⇒1⩽0ithinkis∣sin(x)−sin(y)∣⩽∣x−y∣letf(x,y)=∫xycos(t)dtf(x,y)=[sin(t)]xy=sin(y)−sin(x)∣f(x,y)∣=∣∫xycos(t)dt∣⩽∣∫xy∣cos(t)dt∣∣⩽∣∫xy1dt∣=∣y−x∣⇒∣sin(y)−sin(x)∣⩽∣y−x∣
Commented by Henri Boucatchou last updated on 23/Oct/19
U′re right Sir, using cos or sin only is available.
U′rerightSir,usingcosorsinonlyisavailable.

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