Determine whether the equality is an identity or an equation. If it is an identity, prove it. If it is an equation, solve it for $(0\leq x < 2\pi)$.

$$\sin x \cos x - 1 = \cos x - \sin x$$

Solve each equation. If the equation is an identity or if it has no solution, write identity or no solution. $3(2 t-6)=2(3 t-9)$

Decide whether the equation is an identity. Explain. (An identity is true for any values of the variables.)

$$(x+y)^{3}=x^{3}+3 x^{2} y+3 x y^{2}+y^{3}$$

Verify each identity. Give the domain of validity for each identity. $\sin \theta \tan \theta=\frac{1}{\cos \theta}-\cos \theta$

Only one of the following equations is an identity. Identify the identity and explain your answer. $(x-1)^{2}-1=x(x-2) \quad(x-1)^{2}=x(x-1)$

Every year at the beginning of flu season, many people, including the elderly, get a flu shot to reduce their chances of contracting the flu. One result is that people who do not get a flu shot are less likely to contract the flu.

On the graph that follows, show the effects of this externality by drawing in and labeling any additional curves that are needed and by labeling the efficient quantity and the efficient price of flu shots. Label the area representing deadweight loss in this market.

Why do you think Neruda calls the "constancy that beats in your chest" (literally, his beloved's heartbeat) a "purer motion"? Interpret the remarkable visual image in line $11$.

Activation of the diaphragm and the external intercostal muscles begins the inspiratory process. What effect does contraction of these muscles have on thoracic volume, and how is this accomplished?

$-8(1)=$______ , multiplicative identity property

Recall the quotient identity $\tan x=\frac{\sin x}{\cos x}$. Use the graphs of y = sin x and y = cos x in conjunction with the quotient identity to explain the shape of the graph of y = tan x.

Why is $1$ called the multiplicative identity?

Give an example to show that the stated case can occur. The product of two irrational numbers may be a rational number.

Use a graphing utility to determine whether the proposed identity appears to be an identity. If it does, prove it algebraically. If not, give a counterexample.

$\cos 3 x=3 \cos x$

Use Definition 2.4 .4 or 2.4 .5 to prove that the stated limit is correct.

$$\lim _{x \rightarrow 3} \frac{-1}{(x-3)^2}=-\infty$$