Other trigonometric functions

$$\begin{eqnarray*} \tan(\theta) & := & \frac{\sin(\theta)}{\cos(\theta)} \\ \cot... ...1}{\cos(\theta)} \\ \csc(\theta) & := & \frac{1}{\sin(\theta)} \end{eqnarray*}$$

Derivative of the tangent function

$$\begin{eqnarray*} \frac{d}{dt} (\tan t) & = & \frac{d}{dt} (\frac{\sin t}{\cos t... ... t}{\cos^2 t} \\ & = & \frac{1}{\cos^2 t} \\ & = & \sec^2 t \end{eqnarray*}$$

Derivative of the secant function

$$\begin{eqnarray*} \frac{d}{dt} (\sec t) & = & \frac{d}{dt} (\frac{1}{\cos t}) \\... ... \\ & = & \frac{\sin t}{\cos^2 t} \\ & = & (\tan t)(\sec t) \end{eqnarray*}$$

Differentiating functions built from trigonometric functions

Let $g(t) = \tan(\ln(\sec t))$. Compute $g'(t)$.

Solution

$$\begin{eqnarray*} g'(t) & = & \frac{d}{dx}(\tan x) \vert_{x = \ln( \sec t)} \fra... ...{\sec t} (\tan t) (\sec t) \\ & = & \sec^2(\ln(\sec t)) \tan t \end{eqnarray*}$$