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Precalculus 7th

David Cohen, Theodore B. Lee, David Sklar

Chapter 9

Analytical Trigonometry

Educators


Problem 1

Use the addition formulas for sine and cosine to simplify the expression.
$$\sin \theta \cos 2 \theta+\cos \theta \sin 2 \theta$$

Jose H.
Numerade Educator

Problem 2

Use the addition formulas for sine and cosine to simplify the expression.
$$\sin \frac{\pi}{6} \cos \frac{\pi}{3}+\cos \frac{\pi}{6} \sin \frac{\pi}{3}$$

Jose H.
Numerade Educator

Problem 3

Use the addition formulas for sine and cosine to simplify the expression.
$$\sin 3 \theta \cos \theta-\cos 3 \theta \sin \theta$$

Jose H.
Numerade Educator

Problem 4

Use the addition formulas for sine and cosine to simplify the expression.
$$\sin 110^{\circ} \cos 20^{\circ}-\cos 110^{\circ} \sin 20^{\circ}$$

Jose H.
Numerade Educator

Problem 5

Use the addition formulas for sine and cosine to simplify the expression.
$$\cos 2 u \cos 3 u-\sin 2 u \sin 3 u$$

Jose H.
Numerade Educator

Problem 6

Use the addition formulas for sine and cosine to simplify the expression.
$$\cos 2 u \cos 3 u+\sin 2 u \sin 3 u$$

Jose H.
Numerade Educator

Problem 7

Use the addition formulas for sine and cosine to simplify the expression.
$$\cos \frac{2 \pi}{9} \cos \frac{\pi}{18}+\sin \frac{2 \pi}{9} \sin \frac{\pi}{18}$$

Jose H.
Numerade Educator

Problem 8

Use the addition formulas for sine and cosine to simplify the expression.
$$\cos \frac{3 \pi}{10} \cos \frac{\pi}{5}-\sin \frac{3 \pi}{10} \sin \frac{\pi}{5}$$

Jose H.
Numerade Educator

Problem 9

Use the addition formulas for sine and cosine to simplify the expression.
$$\sin (A+B) \cos A-\cos (A+B) \sin A$$

Jose H.
Numerade Educator

Problem 10

Use the addition formulas for sine and cosine to simplify the expression.
$$\cos (s-t) \cos t-\sin (s-t) \sin t$$

Jose H.
Numerade Educator

Problem 11

Simplify each expression (as in Example 2).
$$\sin \left(\theta-\frac{3 \pi}{2}\right)$$

Jose H.
Numerade Educator

Problem 12

Simplify each expression (as in Example 2).
$$\cos \left(\frac{3 \pi}{2}+\theta\right)$$

Jose H.
Numerade Educator

Problem 13

Simplify each expression (as in Example 2).
$$\cos (\theta+\pi)$$

Jose H.
Numerade Educator

Problem 14

Simplify each expression (as in Example 2).
$$\sin (\theta-\pi)$$

Jose H.
Numerade Educator

Problem 15

Expand $\sin (t+2 \pi)$ using the appropriate addition formula, and check to see that your answer agrees with the fact that sin has period $2 \pi$.

Jose H.
Numerade Educator

Problem 16

Follow the directions in Exercise $15,$ but use $\cos (t+2 \pi)$.

Jose H.
Numerade Educator

Problem 17

Use the formula for $\cos (s+t)$ to compute the exact value of $\cos 75^{\circ}$.

Jose H.
Numerade Educator

Problem 18

Use the formula for $\sin (s-t)$ to compute the exact value of $\sin \frac{\pi}{12}$.

Jose H.
Numerade Educator

Problem 19

Use the formula for $\sin (s+t)$ to find $\sin \frac{7 \pi}{12}$.

Jose H.
Numerade Educator

Problem 20

Determine the exact value of (a) $\sin 105^{\circ}$ and (b) $\cos 105^{\circ}$.

Jose H.
Numerade Educator

Problem 21

Use the addition formulas for sine and cosine to simplify each expression.
$$\sin \left(\frac{\pi}{4}+s\right)-\sin \left(\frac{\pi}{4}-s\right)$$

Jose H.
Numerade Educator

Problem 22

Use the addition formulas for sine and cosine to simplify each expression.
$$\sin \left(t+\frac{\pi}{6}\right)-\sin \left(t-\frac{\pi}{6}\right)$$

Jose H.
Numerade Educator

Problem 23

Use the addition formulas for sine and cosine to simplify each expression.
$$\cos \left(\frac{\pi}{3}-\theta\right)-\cos \left(\frac{\pi}{3}+\theta\right)$$

Jose H.
Numerade Educator

Problem 24

Use the addition formulas for sine and cosine to simplify each expression.
$$\cos \left(\theta-\frac{\pi}{4}\right)+\cos \left(\theta+\frac{\pi}{4}\right)$$

Jose H.
Numerade Educator

Problem 25

Compute the indicated quantity using the following data.
$$\begin{aligned}&\sin \alpha=\frac{12}{13} \quad \text { where } \frac{\pi}{2}<\alpha<\pi\\&\cos \beta=-\frac{3}{5} \quad \text { where } \pi<\beta<\frac{3 \pi}{2}\\&\cos \theta=\frac{7}{25} \quad \text { where }-2 \pi<\theta<-\frac{3 \pi}{2}\end{aligned}$$
(a) $\sin (\alpha+\beta)$
(b) $\cos (\alpha+\beta)$

Jose H.
Numerade Educator

Problem 26

Compute the indicated quantity using the following data.
$$\begin{aligned}&\sin \alpha=\frac{12}{13} \quad \text { where } \frac{\pi}{2}<\alpha<\pi\\&\cos \beta=-\frac{3}{5} \quad \text { where } \pi<\beta<\frac{3 \pi}{2}\\&\cos \theta=\frac{7}{25} \quad \text { where }-2 \pi<\theta<-\frac{3 \pi}{2}\end{aligned}$$
(a) $\sin (\alpha-\beta)$
(b) $\cos (\alpha-\beta)$

Jose H.
Numerade Educator

Problem 27

Compute the indicated quantity using the following data.
$$\begin{aligned}&\sin \alpha=\frac{12}{13} \quad \text { where } \frac{\pi}{2}<\alpha<\pi\\&\cos \beta=-\frac{3}{5} \quad \text { where } \pi<\beta<\frac{3 \pi}{2}\\&\cos \theta=\frac{7}{25} \quad \text { where }-2 \pi<\theta<-\frac{3 \pi}{2}\end{aligned}$$
(a) $\sin (\theta-\beta)$
(b) $\sin (\theta+\beta)$

Jose H.
Numerade Educator

Problem 28

Compute the indicated quantity using the following data.
$$\begin{aligned}&\sin \alpha=\frac{12}{13} \quad \text { where } \frac{\pi}{2}<\alpha<\pi\\&\cos \beta=-\frac{3}{5} \quad \text { where } \pi<\beta<\frac{3 \pi}{2}\\&\cos \theta=\frac{7}{25} \quad \text { where }-2 \pi<\theta<-\frac{3 \pi}{2}\end{aligned}$$
(a) $\cos (\alpha+\theta)$
(b) $\cos (\alpha-\theta)$

Jose H.
Numerade Educator

Problem 29

Suppose that $\sin \theta=1 / 5$ and $0<\theta<\pi / 2$
(a) Compute $\cos \theta$
(b) Compute sin $2 \theta . \quad$

Jose H.
Numerade Educator

Problem 30

Suppose that $\cos \theta=12 / 13$ and $3 \pi / 2<\theta<2 \pi$
(a) Compute $\sin \theta$
(b) Compute cos $2 \theta . \quad$ Hint: $\cos 2 \theta=\cos (\theta+\theta)$

Jose H.
Numerade Educator

Problem 31

Given $\tan \theta=-2 / 3,$ where $\pi / 2<\theta<\pi,$ and $\csc \beta=2$
where $0<\beta<\pi / 2,$ find $\sin (\theta+\beta)$ and $\cos (\beta-\theta)$

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Problem 32

Given sec $s=5 / 4,$ where $\sin s<0,$ and $\cot t=-1,$ where $\pi / 2<t<\pi,$ find $\sin (s-t)$ and $\cos (s+t)$.

Linda H.
Numerade Educator

Problem 33

Prove that each equation is an identity.
$$\sin \left(t+\frac{\pi}{4}\right)=(\sin t+\cos t) / \sqrt{2}$$

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Problem 34

Prove that each equation is an identity.
$$\cos \left(t+\frac{\pi}{4}\right)=(\cos t-\sin t) / \sqrt{2}$$

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Problem 35

Prove that each equation is an identity.
$$\sin \left(t+\frac{\pi}{4}\right)+\cos \left(t+\frac{\pi}{4}\right)=\sqrt{2} \cos t$$

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Problem 36

Prove that each equation is an identity.
$$\sec (\alpha+\beta)=\frac{\sec \alpha \sec \beta}{1-\tan \alpha \tan \beta}$$

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Problem 37

Use the given information to compute $\tan (s+t)$ and $\tan (s-t)$.
$$\tan s=2 \quad \text { and } \quad \tan t=3$$

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Problem 38

Use the given information to compute $\tan (s+t)$ and $\tan (s-t)$.
$$\tan s=1 / 2 \text { and } \tan t=1 / 3$$

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Problem 39

Use the given information to compute $\tan (s+t)$ and $\tan (s-t)$.
$$s=3 \pi / 4 \text { and } \tan t=-4$$

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Problem 40

Use the given information to compute $\tan (s+t)$ and $\tan (s-t)$.
$$s=7 \pi / 4 \text { and } \tan t=-2$$

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Problem 41

Use the addition formulas for tangent to simplify each expression.
$$\frac{\tan t+\tan 2 t}{1-\tan t \tan 2 t}$$

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Problem 42

Use the addition formulas for tangent to simplify each expression.
$$\frac{\tan \frac{\pi}{5}-\tan \frac{\pi}{30}}{1+\tan \frac{\pi}{5} \tan \frac{\pi}{30}}$$

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Problem 43

Use the addition formulas for tangent to simplify each expression.
$$\frac{\tan 70^{\circ}-\tan 10^{\circ}}{1+\tan 70^{\circ} \tan 10^{\circ}}$$

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Problem 44

Use the addition formulas for tangent to simplify each expression.
$$\frac{2 \tan \frac{\pi}{12}}{1-\tan ^{2} \frac{\pi}{12}}$$

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Problem 45

Use the addition formulas for tangent to simplify each expression.
$$\frac{\tan (x-y)+\tan y}{1-\tan (x-y) \tan y}$$

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Problem 46

Use the addition formulas for tangent to simplify each expression.
$$[\tan (\theta+\pi)][\tan (\theta-\pi)]+1$$

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Problem 47

Compute $\tan \frac{7 \pi}{12}$ and rationalize the answer. Hint: $\frac{7 \pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}$.

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Problem 48

Compute tan $15^{\circ}$ using the fact that $15^{\circ}=45^{\circ}-30^{\circ} .$ Then check that your answer is consistent with the result in Example 7.

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Problem 49

Prove that each equation is an identity.
$$\frac{\sin (s+t)}{\cos s \cos t}=\tan s+\tan t$$

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Problem 50

Prove that each equation is an identity.
$$\frac{\cos (s-t)}{\cos s \sin t}=\cot t+\tan s$$

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Problem 51

Prove that each equation is an identity.
$$\cos (A-B)-\cos (A+B)=2 \sin A \sin B$$

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Problem 52

Prove that each equation is an identity.
$$\sin (A-B)+\sin (A+B)=2 \sin A \cos B$$

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Problem 53

Prove that each equation is an identity.
$$\cos (A+B) \cos (A-B)=\cos ^{2} A-\sin ^{2} B$$

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Problem 54

Prove that each equation is an identity.
$$\sin (A+B) \sin (A-B)=\cos ^{2} B-\cos ^{2} A$$

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Problem 55

Prove that each equation is an identity.
$$\cos (\alpha+\beta) \cos \beta+\sin (\alpha+\beta) \sin \beta=\cos \alpha$$

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Problem 56

Prove that each equation is an identity.
$$\cos \left(\theta+\frac{\pi}{4}\right)+\sin \left(\theta-\frac{\pi}{4}\right)=0$$

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Problem 57

Prove that each equation is an identity.
$$\tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta}$$

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Problem 58

Prove that each equation is an identity.
$$\tan \left(\frac{\pi}{4}+\theta\right)-\tan \left(\frac{\pi}{4}-\theta\right)=2 \tan 2 \theta$$

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Problem 59

You are asked to derive expressions for the average rates of change of the functions $\sin x, \cos x,$ and $\tan x$ In each case, assume that the interval is $[x, x+h] .$ (The results are used in calculus in the study of derivatives.)
Let $f(x)=\sin x .$ Show that
$$\frac{\Delta f}{\Delta x}=(\sin x)\left(\frac{\cos h-1}{h}\right)+(\cos x)\left(\frac{\sin h}{h}\right)$$

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Problem 60

You are asked to derive expressions for the average rates of change of the functions $\sin x, \cos x,$ and $\tan x$ In each case, assume that the interval is $[x, x+h] .$ (The results are used in calculus in the study of derivatives.)
Let $g(x)=\cos x .$ Show that
$$\frac{\Delta g}{\Delta x}=(\cos x)\left(\frac{\cos h-1}{h}\right)-(\sin x)\left(\frac{\sin h}{h}\right)$$

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Problem 61

You are asked to derive expressions for the average rates of change of the functions $\sin x, \cos x,$ and $\tan x$ In each case, assume that the interval is $[x, x+h] .$ (The results are used in calculus in the study of derivatives.)
Let $T(x)=\tan x .$ Show that
$$\frac{\Delta T}{\Delta x}=\frac{\tan h}{h} \cdot \frac{\sec ^{2} x}{1-\tan h \tan x}$$

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Problem 62

Let $\theta$ be the acute angle defined by the following figure.
CAN'T COPY THE FIGURE
Use an addition formula and the figure to show that $5 \sin (x+\theta)=4 \sin x+3 \cos x$.

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Problem 63

Let $a$ and $b$ be positive constants, and let $\theta$ be the acute angle (in radian measure) defined by the following figure.
CAN'T COPY THE FIGURE
(a) Use an addition formula and the figure to show that $\sqrt{a^{2}+b^{2}} \sin (x+\theta)=a \sin x+b \cos x$.
(b) Use the result in part (a) to specify the maximum value of the function $f(x)=a \sin x+b \cos x$.

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Problem 64

(a) Use an addition formula to show that $2 \sin \left(x+\frac{\pi}{6}\right)=\cos x+\sqrt{3} \sin x$
(b) Use the result in part (a) to graph the function $f(x)=\cos x+\sqrt{3} \sin x$ for one period.

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Problem 65

(a) Use an addition formula to show that $\sqrt{2} \cos \left(x-\frac{\pi}{4}\right)=\cos x+\sin x$
(b) Use the result in part (a) to graph the function $f(x)=\cos x+\sin x$ for one period.

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Problem 66

Let $A, B,$ and $C$ be the angles of a triangle, so that $A+B+C=\pi$
(a) Show that $\sin (A+B)=\sin C$.
(b) Show that $\cos (A+B)=-\cos C$.
(c) Show that $\tan (A+B)=-\tan C$.

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Problem 67

Suppose that $A, B,$ and $C$ are the angles of a triangle, so that $A+B+C=\pi .$ Show that
$$\cos ^{2} A+\cos ^{2} B+\cos ^{2} C+2 \cos A \cos B \cos C=1$$

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Problem 68

Prove that
$$\frac{\sin (\alpha-\beta)}{\cos \alpha \cos \beta}+\frac{\sin (\beta-\gamma)}{\cos \beta \cos \gamma}+\frac{\sin (\gamma-\alpha)}{\cos \gamma \cos \alpha}=0$$

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Problem 69

Suppose that $a^{2}+b^{2}=1$ and $c^{2}+d^{2}=1 .$ Prove that $|a c+b d| \leq 1 .$ Hint: Let $a=\cos \theta, b=\sin \theta, c=\cos \phi$ and $d=\sin \phi$.

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Problem 70

Simplify the expression.
$$\cos \left(\frac{\pi}{6}+t\right) \cos \left(\frac{\pi}{6}-t\right)-\sin \left(\frac{\pi}{6}+t\right) \sin \left(\frac{\pi}{6}-t\right)$$

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Problem 71

Simplify the expression.
$$\sin \left(\frac{\pi}{3}-t\right) \cos \left(\frac{\pi}{3}+t\right)+\cos \left(\frac{\pi}{3}-t\right) \sin \left(\frac{\pi}{3}+t\right)$$

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Problem 72

Simplify the expression.
$$\frac{\tan (A+2 B)-\tan (A-2 B)}{1+\tan (A+2 B) \tan (A-2 B)}$$

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Problem 73

If $\alpha+\beta=\pi / 4,$ show that $(1+\tan \alpha)(1+\tan \beta)=2$.

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Problem 74

Exercises 74 and 75 outline simple geometric derivations of the formulas for sin $(\alpha+\beta)$ and $\cos (\alpha+\beta)$ in the case in which $\alpha$ and $\beta$ are acute angles, with $\alpha+\beta<90^{\circ} .$ The exercises rely on the accompanying figures, which are constructed as follows.
Begin, in Figure $A,$ with $\alpha=\angle G A D, \beta=\angle H A G,$ and $A H=1$ Then, from $H,$ draw perpendiculars to $A D$ and to $A G,$ as shown in Figure $B$. Finally, draw $\overline{F E} \perp \overline{B H}$ and $\overline{F C} \perp \overline{A D}$.
a.CAN'T COPY THE FIGURE
b.CAN'T COPY THE FIGURE
Formula for $\sin (\alpha+\beta) . \quad$ Supply the reasons or steps behind each statement.
(a) $B H=\sin (\alpha+\beta)$
(b) $F H=\sin \beta$
(c) $\angle B H F=\alpha$
(d) $E H=\cos \alpha \sin \beta \quad$ Hint: Use $\triangle E F H$ and the result in part (b)
(e) $A F=\cos \beta$
(f) $C F=\sin \alpha \cos \beta$
(g) $\sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta$

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Problem 75

Exercises 74 and 75 outline simple geometric derivations of the formulas for sin $(\alpha+\beta)$ and $\cos (\alpha+\beta)$ in the case in which $\alpha$ and $\beta$ are acute angles, with $\alpha+\beta<90^{\circ} .$ The exercises rely on the accompanying figures, which are constructed as follows.
Begin, in Figure $A,$ with $\alpha=\angle G A D, \beta=\angle H A G,$ and $A H=1$ Then, from $H,$ draw perpendiculars to $A D$ and to $A G,$ as shown in Figure $B$. Finally, draw $\overline{F E} \perp \overline{B H}$ and $\overline{F C} \perp \overline{A D}$.
a.CAN'T COPY THE FIGURE
b.CAN'T COPY THE FIGURE
Formula for $\cos (\alpha+\beta) .$ Supply the reasons or steps behind each statement.
(a) $\cos (\alpha+\beta)=A B$
(b) $A C=\cos \alpha \cos \beta$
(c) $E F=\sin \alpha \sin \beta$
(d) $\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta$

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Problem 76

Let $S$ and $C$ be two functions. Assume that the domain for both $S$ and $C$ is the set of all real numbers and that $S$ and $C$ satisfy the following two identities.
$$\begin{aligned}&1)S(x-y)=S(x) C(y)-C(x) S(y)\\&2)C(x-y)=C(x) C(y)+S(x) S(y)\end{aligned}$$
Also, suppose that the function S is not identically zero. That is,
$3)S(x) \neq 0 \quad$ for at least one real number $x$
(a) Show that $S(0)=0 .$
(b) Show that $C(0)=1 .$
(c) Explain (in complete sentences) why it was necessary to use condition (3) in the work for part (b).
(d) Prove the identity $[C(x)]^{2}+[S(x)]^{2}=1$ Hint: In identity $(2),$ let $y=x$
(e) Show that $C$ is an even function and $S$ is an odd function. That is, prove the identities $C(-x)=C(x)$ and $S(-x)=-S(x)$

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Problem 77

Prove the identities.
$$\frac{\sin (A+B)}{\sin (A-B)}=\frac{\tan A+\tan B}{\tan A-\tan B}$$

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Problem 78

Prove the identities.
$$\frac{\cos (A+B)}{\cos (A-B)}=\frac{1-\tan A \tan B}{1+\tan A \tan B}$$

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Problem 79

Prove the identities.
$$\cot (A+B)=\frac{\cot A \cot B-1}{\cot A+\cot B}$$

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Problem 80

Prove the identities.
$$\cot (A-B)=\frac{\cot A \cot B+1}{\cot B-\cot A}$$

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Problem 81

Let $f(t)=\cos ^{2} t+\cos ^{2}\left(t+\frac{2 \pi}{3}\right)+\cos ^{2}\left(t-\frac{2 \pi}{3}\right)$
(a) Complete the table. (Use a calculator.)
CAN'T COPY THE FIGURE
(b) On the basis of your results in part (a), make a conjecture about the function $f$. Prove that your conjecture is correct.

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Problem 82

(a) Use your calculator to check that $\tan 50^{\circ}-\tan 40^{\circ}=2 \tan 10^{\circ}$
(b) Part (a) is a specific example of a more general identity. State the identity and prove it.

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Problem 83

If $\tan B=\frac{n \sin A \cos A}{1-n \sin ^{2} A},$ show that
$\tan (A-B)=(1-n) \tan A$.

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Problem 84

If triangle $A B C$ is not a right triangle, and $\cos A=\cos B \cos C,$ show that $\tan B \tan C=2$.

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Problem 85

(a) The angles of a triangle are $A=20^{\circ}, B=50^{\circ},$ and $C=110^{\circ} .$ Use your calculator to compute the sum $\tan A+\tan B+\tan C$ and then the product
$\tan A \tan B \tan C .$ What do you observe?
(b) The angles of a triangle are $\alpha=\pi / 10, \beta=3 \pi / 10$ and $\gamma=3 \pi / 5 .$ Use your calculator to compute $\tan \alpha+\tan \beta+\tan \gamma$ and $\tan \alpha \tan \beta \tan \gamma$
(c) If triangle $A B C$ is not a right triangle, prove that $\tan A+\tan B+\tan C=\tan A \tan B \tan C$.

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