## The Derivatives of Trigonometric Functions

Our starting point is the following limit:

To see why, it is enough to rewrite the expression involving the cosine as

But , so we have

This limit equals and thus .

In fact, we may use these limits to find the derivative of and at any point

*x*=

*a*. Indeed, using the addition formula for the sine function, we have

So

which implies

So we have proved that exists and .Similarly, we obtain that exists and that .

Since , , , and are all quotients of the functions and , we can compute their derivatives with the help of the quotient rule:

It is quite interesting to see the close relationship between and (and also between and ).

From the above results we get

These two results are very useful in solving some differential equations.

**Example 1.**Let . Using the double angle formula for the sine function, we can rewrite

So using the product rule, we get

which implies, using trigonometric identities,

**Exercise 1.**Find the equations of the tangent line and the normal line to the graph of at the point .

**Answer.**First we need to find the derivative of

*f*(

*x*) so we can get the slope of the tangent line and the normal line. We have

So we have

knowing that and . Note that

So the equation of the tangent line at the point is

the slope of the normal line to the graph at the point is

which helps us find the equation of the normal line as

**Exercise 2.**Find the

*x*-coordinates of all points on the graph of in the interval at which the tangent line is horizontal.

**Answer.**The points (

*x*,

*f*(

*x*)) at which the tangent line is horizontal are the ones for which

*f*'(

*x*) = 0. Let us first find

*f*'(

*x*). We have

So we have to solve which gives . We get