**1. Introduction.** So we have investigated several useful trig limits, lets consider the bigger problem: differentiating trigonometric functions.

We will use the following properties without proving them:

(1a)

(1b)

(1c)

(1d)

Where we have be arbitrary real numbers.

**2. Derivative of Cosine.** We see from (1a) that

(2)

We divide both sides by obtaining

(3).

Taking the limit as , and recalling our useful trigonometric limits, we have the definition of the derivative for cosine

(4).

**3. Derivative of Sine.** Likewise, using Equation (1b), we find

(5).

Dividing through by yields

(6)

Again, taking the limit yields the derivative

(7)

Using our beloved trig limits produces:

(8)

Thus we obtain

(9).

**Exercise 1.** Prove .

**Exercise 2.** Recall the secant function is . What is its derivative?

**Exercise 3.** Recall the cosecant function is . What is its derivative?

**Exercise 4.** What is the derivative of with respect to ?

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