Last Activity: 16 Years ago
I am providing you the step wise solution for your problem:
Step 1. Function cos x is continuous and differentiable for all real numbers. Use the mean value theorem, using 2 real numbers a and b to write
(cos x) ' = [cos a - cos b] / [a - b]
step 2. Take the absolute value of both sides
| (cos x) ' | = | [cos a - cos b] / [a - b] |
| (cos x) ' | < = 1
step 3. Which gives
| [cos a - cos b] / [a - b] | <= 1
step 4. But
| [cos a - cos b] / [a - b] | = |cos a - cos b| / |a - b|
step 5. When combined with the above gives
|cos a - cos b| / |a - b| <= 1
step 6. Multiply both sides by |a - b| to obtain
|cos a - cos b| <= |a - b|
Thanks
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