1. ## Maximum value

find the maximum value of
(cos(cos x))^2 +(sin(sin x))^2

find the maximum value of
(cos(cos x))^2 +(sin(sin x))^2
$\displaystyle u = \cos{x}$ , $\displaystyle v = \sin{x}$

$\displaystyle y = \cos^2(u) + \sin^2(v)$

$\displaystyle \dfrac{dy}{dx} = -2\cos(u)\sin(u) \cdot \dfrac{du}{dx} + 2\sin(v)\cos(v) \cdot \dfrac{dv}{dx}$

take it from here?

3. Originally Posted by skeeter
$\displaystyle u = \cos{x}$ , $\displaystyle v = \sin{x}$

$\displaystyle y = \cos^2(u) + \sin^2(v)$

$\displaystyle \dfrac{dy}{dx} = -2\cos(u)\sin(u) \cdot \dfrac{du}{dx} + 2\sin(v)\cos(v) \cdot \dfrac{dv}{dx}$

take it from here?
not getting

not getting
Are you even trying? You were told that u= cos(x). What is du/dx? You were told that v= sin(x). What is dv/dx?

find the maximum value of (cos(cos x))^2 +(sin(sin x))^2

The function

$\displaystyle f(x)= (\;\cos (\cos x)\^2+(\;\sin (\sin x)\^2$

is even and periodic with period $\displaystyle \pi$ so, we only need to find the maximum on $\displaystyle [0,\pi/2]$ .

The values of $\displaystyle f$ at the endpoints of $\displaystyle [0,\pi/2]$ are:

$\displaystyle f(0)=\cos^2 1<\sin^2 1=f(\pi/2)$

Prove that there are no singular points in $\displaystyle (0,\pi/2)$ so, the absolute maximun of $\displaystyle f$ is $\displaystyle \sin^2 1$ and the absolute minimum $\displaystyle \cos^2 1$ .