Using algebra similar to how we found the Power Rule, the shortcuts for taking the derivatives of trigonometric functions turn out to be:

$$ {\operatorname{d}\!y \over \operatorname{d}\!x} [  \sin x  ]    =   \cos x $$

$$ {\operatorname{d}\!y \over \operatorname{d}\!x} [  \cos x  ]    =   -\sin x $$

$$ {\operatorname{d}\!y \over \operatorname{d}\!x} [  \tan x  ]    =   {\sec^2} x $$

$$ {\operatorname{d}\!y \over \operatorname{d}\!x} [  \cot x  ]    =   -{\csc^2} x $$

$$ {\operatorname{d}\!y \over \operatorname{d}\!x} [  \sec x  ]    =   \sec x   \tan x $$

$$ {\operatorname{d}\!y \over \operatorname{d}\!x} [  \csc x  ]    =   -\csc x   \cot x $$