Please solve the following problem by using Mean Valu theorem only

For any Two real Numbers a and b | cos a - cos b | <= | a - b | .

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|