Verify formulas for the derivatives of some trig functions

Verify the following formulas:

Let $f(x) = \tan x = \frac{\sin x}{\cos x}$ . We use the quotient rule and the derivatives of sine and cosine to compute,
$\begin{align*} f'(x) &= \frac{(\sin x)'(\cos x) - (\sin x)(\cos x)'}{\cos^2 x} \\ &= \frac{\cos^2 x + \sin^2 x}{\cos^2 x} \\ &= \frac{1}{\cos^2 x} \\ &= \sec^2 x. \qquad \blacksquare \end{align*}$
Let $f(x) = \cot x = \frac{\cos x}{\sin x}$ . Using the quotient rule and derivatives of sine and cosine again,
$\begin{align*} f'(x) &= \frac{(\cos x)'(\sin x) - (\cos x)(\sin x)'}{\sin^2 x} \\ &= \frac{-\sin^2 x - \cos^2 x}{\sin^2 x} \\ &= \frac{-1}{\sin^2 x} \\ &= -\csc^2 x. \qquad \blacksquare \end{align*}$
Let $f(x) = \sec x = \frac{1}{\cos x}$ . Again, we compute,
$\begin{align*} f'(x) &= \frac{\sin x}{\cos^2 x} \\ &= \left( \frac{\sin x}{\cos x} \right) \left( \frac{1}{\cos x} \right)\\ &= \tan x \sec x. \qquad \blacksquare \end{align*}$
Let $f(x) = \csc x = \frac{1}{\sin x}$ . Then,
$\begin{align*} f'(x) &= \frac{-\cos x}{\sin^2 x} \\ &= \left( \frac{-\cos x}{\sin x} \right) \left( \frac{1}{\sin x} \right) \\ &= - \cot x \csc x. \qquad \blacksquare \end{align*}$