In this exercise, verify the identity.

$$\cos ^2 \frac{\theta}{2}=\frac{\sec \theta+1}{2 \sec \theta}$$

Write the products from reaction of the following reagents with p-dimethylbenzene (also called p-xylene): (a) $\mathrm{Br}_2, \mathrm{FeBr}_3$ (b) $\mathrm{HNO}_3, \mathrm{H}_2 \mathrm{SO}_4$

For the following exercise,

Find the exact value of each trigonometric function.

$$\cos \frac{\pi}{6}$$

How many electrons does it take to have a total charge of - 1 C?

In this exercise, find the exact value of each expression. Do not use a calculator.

$$\cos ^{-1} 1$$

In this exercise, evaluate the function for the given value of $x$.

$$g(x)=x^4-32 x^2+256 ; x=-4$$

For the following exercise,

Find the exact value.

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

Solve the following exercise. Find the value of the expression given below.

$$22.81 \times 10^{-6}$$

In this exercise, use a calculator to find the following value of theta to the nearest degree:

$$\theta=\arccos 0.9$$

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

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

Quinoline and isoquinoline react with organometallic reagents exactly as pyridine does. Give the products of their reaction with 2-propenylmagnesium bromide (allylmagnesium bromide).

For the following exercise,

Find the exact value of each trigonometric function.

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

Determine if the statement is true. If it is false, explain why and correct it.

$$\frac{d}{dx}[\sec x^2]=2x\sec x\tan x$$

Verify that each equation is an identity.

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