In order to find the derivatives of the trigonometric functions the definition of the derivative must be used. Each of the following cases uses the form

$$ f’\left(x \right) = \lim_{h \rightarrow 0} \frac{f \left( x + h \right) – f \left( x \right)}{h} $$

Let \( f \left(x \right) = \sin \left( x \right) \)

$$ \lim_{h \rightarrow 0} \frac{ \sin \left( x + h \right) – \sin \left( x \right)}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin \left( x \right) \cos \left( h \right) + \sin \left( h \right) \cos \left( x \right) – \sin \left( x \right)}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin \left( x \right) \left( \cos h – 1 \right) + \cos \left( x \right) \sin \left( h \right)}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin \left( x \right) \left( \cos h – 1 \right)}{h} + \frac{\cos \left( x \right) \sin \left( h \right)}{h} = \cos \left( x \right) $$

Let \( f \left(x \right) = \cos x \)

$$ \lim_{h \rightarrow 0} \frac{ \cos \left( x + h \right) – \cos x}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \cos x \cos h – \sin h \sin x – \cos x}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \cos x \left( \cos h – 1 \right) – \sin h \sin x}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \cos x \left( \cos h – 1 \right)}{h} – \frac{\sin x \sin h}{h} = – \sin x $$

Let \( f \left(x \right) = \tan x \)

$$\lim_{h \rightarrow 0} \frac{ \tan \left( x + h \right) – \tan x}{h}$$

$$\lim_{h \rightarrow 0} \frac{ \frac{\sin \left(x + h \right)}{\cos \left(x + h \right)} – \frac{ \sin x}{\cos x}}{h}$$

$$ \lim_{h \rightarrow 0} \frac{\sin \left(x + h \right) \cos x – \sin x \left( \cos \left( x + h \right) \right)}{h \cos \left(x + h \right) \cos x} $$

$$ \lim_{h \rightarrow 0} \frac{\left( \sin x \cos h + \sin h \cos x \right) \cos x – \sin x \left( \cos x \cos h – \sin x \sin h \right)}{h \cos \left(x + h \right) \cos x} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin h \cos^2 x}{h \cos \left(x + h \right) \cos x} + \frac{\sin^2 x \sin h}{h \cos \left(x + h \right) \cos x}= 1 + \tan^2 x = \sec^2 x $$

Let \( f \left(x \right) = \csc x \)

$$ \lim_{h \rightarrow 0} \frac{ \csc \left( x + h \right) – \csc x}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \frac{1}{\sin \left(x + h \right)} – \frac{1}{\sin x}}{h} $$

$$ \lim_{h \rightarrow 0} \frac{\sin x – \sin \left( x + h \right)}{h \sin \left(x + h \right) \sin x} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin x – \sin x \cos h – \sin h \cos x}{h \sin \left(x + h \right) \sin x} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin x \left( 1 – \cos h \right)}{h \sin \left(x + h \right) \sin x} – \frac{\sin h \cos x}{h \sin \left(x + h \right) \sin x}= – \cot x \csc x $$

Let \( f \left(x \right) = \sec x \)

$$\lim_{h \rightarrow 0} \frac{ \sec \left( x + h \right) – \sec x}{h}$$

$$\lim_{h \rightarrow 0} \frac{ \frac{1}{\cos \left(x + h \right)} – \frac{1}{\cos x}}{h}$$

$$\lim_{h \rightarrow 0} \frac{\cos x – \cos \left( x + h \right)}{h \cos \left(x + h \right) \cos x}$$

$$\lim_{h \rightarrow 0} \frac{ \cos x – \cos x \cos h + \sin h \sin x}{h \cos \left(x + h \right) \cos x}$$

$$\lim_{h \rightarrow 0} \frac{ \cos x \left( 1 – \cos h \right)}{h \cos \left(x + h \right) \cos x} + \frac{\sin h \sin x}{h \cos \left(x + h \right) \cos x}= \tan x \sec x$$