The trigonometric functions are often appeared in differential calculus. So, it is important to learn the differentiation of all trigonometric functions because they are used as formulas in calculus to deal trigonometric functions easily.

The derivative of $\sin{x}$ with respect $x$ is equal to $\cos{x}$.

$\dfrac{d}{dx}{\sin{x}} \,=\, \cos{x}$

The derivative of $\cos{x}$ with respect $x$ is equal to negative $\sin{x}$.

$\dfrac{d}{dx}{\cos{x}} \,=\, -\sin{x}$

The derivative of $\tan{x}$ with respect $x$ is equal to square of $\sec{x}$.

$\dfrac{d}{dx}{\tan{x}} \,=\, \sec^2{x}$

The derivative of $\cot{x}$ with respect $x$ is equal to negative of square of $\csc{x}$.

$\dfrac{d}{dx}{\cot{x}} \,=\, -\csc^2{x}$ (or) $-\operatorname{cosec}^2{x}$

The derivative of $\sec{x}$ with respect $x$ is equal to product of $\sec{x}$ and $\tan{x}$.

$\dfrac{d}{dx}{\sec{x}} \,=\, \sec{x}.\tan{x}$

The derivative of $\csc{x}$ with respect $x$ is equal to negative of product of $\csc{x}$ and $\cot{x}$.

$\dfrac{d}{dx}{\csc{x}} \,=\, -\csc{x}.\cot{x}$ (or) $-\operatorname{cosec}{x}.\cot{x}$

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