In differential calculus, there are six derivative formulas to find the differentiation of the trigonometric functions. Each derivative rule is given here with mathematical proof.

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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