Integrate

$$\int \sec x \tan x d x$$

Verify that each equation is an identity.

$$\cos 2 \theta=\frac{2-\sec ^2 \theta}{\sec ^2 \theta}$$

For the following exercise,

Find the exact value of each trigonometric function.

$$\cos \frac{\pi}{2}$$

In this exercise, use a calculator to find the value of the trigonometric function to four decimal places.

$$\csc 1$$

For the following exercise,

Find the exact value.

$$\sec \left(\frac{3 \pi}{8}\right)$$

Solve each equation for x, where x in [0, 2pi].

a) sec x csc x - 2 csc x = 0

b) 3 sec ^2 x - 4 = 0

c) 2 sin x sec x - 2 square root of 3 sin x = 0

d) 2 cot x + sec^2 x = 0

e) cot x csc^2 x = 2 cot x

f) 3 tan^3 x - tan x = 0

In this exercise, use a calculator to find the value of the trigonometric function to four decimal places.

$$\tan 3.4$$

For the following exercise,

Let $f(x)=\frac{3}{5} \cos (6 x)$.

What is the smallest possible value for $f(x)$?

Rewrite $\dfrac{1}{n}\sec^2\left(1 + \dfrac{1}{n}\right) + \dfrac{1}{n}\sec^2\left(1 + \dfrac{2}{n}\right) + \dfrac{1}{n}\sec^2\left(1 + \dfrac{3}{n}\right) + \ldots + \dfrac{1}{n}\sec^2\left(1 + \dfrac{n}{n}\right)$ using sigma notation. Do not evaluate.

In this exercise, find the exact value of the following trigonometric function:

$$\cos \frac{2 \pi}{3}$$

What are the products of reaction of (E)-3-methyl-3-hexene with each of the reagents in the indicated problem?

For the following exercise,

Find the exact value of each trigonometric function.

$$\sin \frac{\pi}{2}$$

Find $y^{\prime}$.

y=ln sec x

Show how you would synthesize 2-phenylhex-3-yn-2-ol, starting with acetophenone $\left(\mathrm{PhCOCH}_{3}\right)$ and any other reagents you need. (“2-ol” means there is an OH group on C2.)