The Cosine Function

Consider the function \(f(x)=\cos x\).   What does the derivative look like?   Before that calculation can begin, some special limits must be observed.   Using numerical evidence from tables, it can be shown that:

Now the derivative of the sine function can begin.   Using the definition of derivative and appropriate trig identities:

Apply the definition of derivative:	\(\lim_{h\rightarrow0}\large{\left(\frac{\cos(x+h)-\cos x}{h}\right)}\)
Apply the Angle-Sum identity:	\(\begin{array}{l} \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta \\ \\ \implies \lim_{h\rightarrow0}\large{\left(\frac{\cos x\cos h-\sin x\sin h-\cos x}{h}\right)} \end{array}\)
Collect and separate like terms involving \(\sin x\) and \(\cos x\):	\(\implies \lim_{h\rightarrow0}\large{\left(\frac{\cos x\cos h-\cos x}{h}-\frac{\sin x\sin h}{h}\right)}\)
Apply Limit Laws and view \(\sin x\) and \(\cos x\) as constants:	\(\implies \cos x\cdot\lim_{h\rightarrow0}\large{\left(\frac{\cos h-1}{h}\right)}\normalsize{-\sin x\cdot\lim_{h\rightarrow0}}\large{\left(\frac{\sin h}{h}\right)}\)
Use the Special Limit values from above:	\(\implies \cos x(0)-\sin x(1)\implies-\sin x\)
Therefore, the derivative is \(f^\prime(x)=-\sin x\).

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Other Trigonometric Functions

Using the ideas above, the same arguement gives \(\frac{d}{dx}\left(\sin x\right)=\cos x\).   To establish the derivatives of the remaining trigonometric functions, a combination of the derivatives of \(\sin\) and \(\cos\), trig identities and the Product or Quotient rules are used.   The following video discusses the derivative for the tangent function.

Video - Tangent Derivative

The remaining derivatives can be found using very similar arguement presented here.   I would encourage you to try and work through them with details, but the following table summarizes all the trigonometric function derivatives:

Trigonometric Derivatives
\(\frac{d}{dx}\left(\sin x\right)=\cos x\)	\(\frac{d}{dx}\left(\cos x\right)=-\sin x\)
\(\frac{d}{dx}\left(\tan x\right)=\sec^2 x\)	\(\frac{d}{dx}\left(\cot x\right)=-\csc^2 x\)
\(\frac{d}{dx}\left(\sec x\right)=\sec x\cdot\tan x\)	\(\frac{d}{dx}\left(\csc x\right)=-\csc x\cdot\cot x\)
NOTE:   The derivatives of a trig function that start with a "co" have a negative sign:   \(\cos\), \(\cot\) and \(\csc\).

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