Real analysis

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29.2 Prove that |cos x - cos y| ≤ |x - y| for all x, y in R.

Solution:

In order to prove the above statement, we need to prove first some trigonometric basic formulas.

1) Let's prove the identity:
( 1)
Proof:
We start from the basic formulas:
( 2)
( 3)
By subtracting first equality from the second one, we will have
( 4)
We denote now
( 5)
By adding the 2 equations, one ...