Derivatives of Trig Functions

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Derivatives of Trig Functions

Two important Limits
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Tangent, Cotangent, Secant, and Cosecant
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Derivatives of Tangent, Cotangent, Secant, and Cosecant

We can get the derivatives of the other four trig functions by applying the quotient rule to sine and cosine. For instance,

$\begin{eqnarray*} \frac{d }{dx}\big( \tan(x)\big) & = & \left (\frac{\sin(x)}{\cos(x)} \right )' \cr & = & \frac{\cos(x) (\sin(x))' - \sin(x) (\cos(x))'}{\cos^2(x)} \cr & = & \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} \cr & = & \frac{1}{\cos^2(x)} \cr &=& \sec^2(x). \end{eqnarray*}$

The final results are:

$\frac{d}{dx}\tan(x) = \sec^2(x), \qquad \frac{d}{dx}\sec(x) =\sec(x) \tan(x),\\\frac{d}{dx}\cot(x)=-\csc^2(x), \qquad \frac{d}{dx}\csc(x) = -\csc(x) \cot(x).$