Composition of trigonometric functions, mean value theorem

[zbot1]

Homework Statement

how to show using MVT that cos(cos x) is a contraction.

Homework Equations

| d/dx (cos(cos x)) | = | sin(cos x) sin(x) | < sin 1 < 1

The Attempt at a Solution

Using that relation, the original problem is easily solved. My question is, how do we know:

| sin(cos x) sin(x) | < sin 1 ?

-zbot1

 

[Dick]

Homework Statement

how to show using MVT that cos(cos x) is a contraction.

Homework Equations

| d/dx (cos(cos x)) | = | sin(cos x) sin(x) | < sin 1 < 1

The Attempt at a Solution

Using that relation, the original problem is easily solved. My question is, how do we know:

| sin(cos x) sin(x) | < sin 1 ?

-zbot1

-1<=cos(x)<=1. What does that make the range of values for sin(cos(x))?

 

[zbot1]

Thanks!