The derivatives of trigonometric functions
It is possible to find the derivative of trigonometric functions.
Here is a list of the standard forms that you need to know:

d (sin x) = cos x
dx

d (cos x) = - sin x
dx

d (sec x) = sec x tan x
dx

d (cosec x) = -cosec x cot x
dx

d (tan x) = sec²x
dx

d (cot x) = -cosec²x
dx

One condition upon these results is that x must be measured in radians.

Applying the Chain Rule
The chain rule is used to differentiate harder trigonometric functions.

Example:
Differentiate cos³x with respect to x.
Let y = cos³x
Let u = cos x
therefore y = u³
dy = 3u²
du

du = -sin x
dx

dy = du × dy
dx    dx    du

= -sin x × 3u²

= -sin x × 3cos²x

= -3cos²x sin x