There are six trigonometric functions in trigonometry. Each trigonometric function often involves in differential calculus. So, it is very important to learn the derivative of each trigonometric ratio. The following is the list of derivatives of trigonometric functions and they are used as formulas in differential calculus.
The derivative of $\sin{x}$ with respect $x$ is equal to $cos{x}$.
$\large \dfrac{d}{dx} \, \sin{x} = \cos{x}$
The derivative of $\cos{x}$ with respect $x$ is equal to negative $\sin{x}$.
$\large \dfrac{d}{dx} \, \cos{x} = -\sin{x}$
The derivative of $\tan{x}$ with respect $x$ is equal to square of $sec{x}$.
$\large \dfrac{d}{dx} \, \tan{x} = \sec^2{x}$
The derivative of $\cot{x}$ with respect $x$ is equal to negative of square of $\csc{x}$.
$\large \dfrac{d}{dx} \, \cot{x} = -\csc^2{x}$
The derivative of $\sec{x}$ with respect $x$ is equal to product of $sec{x}$ and $tan{x}$.
$\large \dfrac{d}{dx} \, \sec{x} = \sec{x} \tan{x}$
The derivative of $\csc{x}$ with respect $x$ is equal to negative of product of $\csc{x}$ and $\cot{x}$.
$\large \dfrac{d}{dx} \, \csc{x} = -\csc{x} \cot{x}$
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