Stewart Precalc 6e Section 7.3: Double-Angle, Half-Angle, and Product-Sum Formulas

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Stewart Precalc 6e Section 7.3: Double-Angle, Half-Angle, and Product-Sum Formulas 0/120
1 Concept · Level 1
If we know the values of \(\sin x\) and \(\cos x\), we can find the value of \(\sin 2x\) by using the _______ Formula for Sine. State the formula: \(\sin 2x =\) _______.
2 Concept · Level 1
If we know the value of \(\cos x\) and the quadrant in which \(\dfrac{x}{2}\) lies, we can find the value of \(\sin\left(\dfrac{x}{2}\right)\) by using the _______ Formula for Sine. State the formula: \(\sin\left(\dfrac{x}{2}\right) =\) _______.
3 Skill - Double-Angle · Level 2
Find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) from the given information. \(\sin x = \dfrac{5}{13}\), \(x\) in Quadrant I.
4 Skill - Double-Angle · Level 2
Find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) from the given information. \(\tan x = -\dfrac{4}{3}\), \(x\) in Quadrant II.
5 Skill - Double-Angle · Level 2
Find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) from the given information. \(\cos x = \dfrac{4}{5}\), \(\csc x < 0\).
6 Skill - Double-Angle · Level 2
Find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) from the given information. \(\csc x = 4\), \(\tan x < 0\).
7 Skill - Double-Angle · Level 2
Find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) from the given information. \(\sin x = -\dfrac{3}{5}\), \(x\) in Quadrant III.
8 Skill - Double-Angle · Level 2
Find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) from the given information. \(\sec x = 2\), \(x\) in Quadrant IV.
9 Skill - Double-Angle · Level 2
Find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) from the given information. \(\tan x = -\dfrac{1}{3}\), \(\cos x > 0\).
10 Skill - Double-Angle · Level 2
Find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) from the given information. \(\cot x = \dfrac{2}{3}\), \(\sin x > 0\).
11 Skill - Lowering Powers · Level 3
Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine, as in Example 4. \(\sin^4 x\).
12 Skill - Lowering Powers · Level 3
Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine. \(\cos^4 x\).
13 Skill - Lowering Powers · Level 3
Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine. \(\cos^2 x \sin^4 x\).
14 Skill - Lowering Powers · Level 3
Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine. \(\cos^4 x \sin^2 x\).
15 Skill - Lowering Powers · Level 3
Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine. \(\cos^4 x \sin^4 x\).
16 Skill - Lowering Powers · Level 3
Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine. \(\cos^6 x\).
17 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\sin 15^{\circ}\).
18 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\tan 15^{\circ}\).
19 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\tan 22.5^{\circ}\).
20 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\sin 75^{\circ}\).
21 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\cos 165^{\circ}\).
22 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\cos 112.5^{\circ}\).
23 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\tan\left(\dfrac{\pi}{8}\right)\).
24 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\cos\left(3 \dfrac{\pi}{8}\right)\).
25 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\cos\left(\dfrac{\pi}{12}\right)\).
26 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\tan\left(5 \dfrac{\pi}{12}\right)\).
27 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\sin\left(9 \dfrac{\pi}{8}\right)\).
28 Skill - Half-Angle Exact Value · Level 2
Use an appropriate Half-Angle Formula to find the exact value of the expression. \(\sin\left(11 \dfrac{\pi}{12}\right)\).
29 Skill - Simplify · Level 2
Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(2 \sin 18^{\circ} \cos 18^{\circ}\). (b) \(2 \sin 3 \theta \cos 3 \theta\).
30 Skill - Simplify · Level 2
Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\dfrac{2 \tan 7^{\circ}}{1 - \tan^2 7^{\circ}}\). (b) \(\dfrac{2 \tan 7 \theta}{1 - \tan^2 7 \theta}\).
31 Skill - Simplify · Level 2
Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\cos^2 34^{\circ} - \sin^2 34^{\circ}\). (b) \(\cos^2 5 \theta - \sin^2 5 \theta\).
32 Skill - Simplify · Level 2
Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\cos^2\left(\dfrac{\theta}{2}\right) - \sin^2\left(\dfrac{\theta}{2}\right)\). (b) \(2 \sin\left(\dfrac{\theta}{2}\right) \cos\left(\dfrac{\theta}{2}\right)\).
33 Skill - Simplify · Level 2
Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\dfrac{\sin 8^{\circ}}{1 + \cos 8^{\circ}}\). (b) \(\dfrac{1 - \cos 4 \theta}{\sin 4 \theta}\).
34 Skill - Simplify · Level 2
Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) \(\sqrt{\dfrac{1 - \cos 30^{\circ}}{2}}\). (b) \(\sqrt{\dfrac{1 - \cos 8 \theta}{2}}\).
35 Proof · Level 3
Use the Addition Formula for Sine to prove the Double-Angle Formula for Sine.
36 Proof · Level 3
Use the Addition Formula for Tangent to prove the Double-Angle Formula for Tangent.
37 Skill - Half-Angle · Level 2
Find \(\sin\left(\dfrac{x}{2}\right)\), \(\cos\left(\dfrac{x}{2}\right)\), and \(\tan\left(\dfrac{x}{2}\right)\) from the given information. \(\sin x = \dfrac{3}{5}\), \(0^{\circ} < x < 90^{\circ}\).
38 Skill - Half-Angle · Level 2
Find \(\sin\left(\dfrac{x}{2}\right)\), \(\cos\left(\dfrac{x}{2}\right)\), and \(\tan\left(\dfrac{x}{2}\right)\) from the given information. \(\cos x = -\dfrac{4}{5}\), \(180^{\circ} < x < 270^{\circ}\).
39 Skill - Half-Angle · Level 2
Find \(\sin\left(\dfrac{x}{2}\right)\), \(\cos\left(\dfrac{x}{2}\right)\), and \(\tan\left(\dfrac{x}{2}\right)\) from the given information. \(\csc x = 3\), \(90^{\circ} < x < 180^{\circ}\).
40 Skill - Half-Angle · Level 2
Find \(\sin\left(\dfrac{x}{2}\right)\), \(\cos\left(\dfrac{x}{2}\right)\), and \(\tan\left(\dfrac{x}{2}\right)\) from the given information. \(\tan x = 1\), \(0^{\circ} < x < 90^{\circ}\).
41 Skill - Half-Angle · Level 2
Find \(\sin\left(\dfrac{x}{2}\right)\), \(\cos\left(\dfrac{x}{2}\right)\), and \(\tan\left(\dfrac{x}{2}\right)\) from the given information. \(\sec x = \dfrac{3}{2}\), \(270^{\circ} < x < 360^{\circ}\).
42 Skill - Half-Angle · Level 2
Find \(\sin\left(\dfrac{x}{2}\right)\), \(\cos\left(\dfrac{x}{2}\right)\), and \(\tan\left(\dfrac{x}{2}\right)\) from the given information. \(\cot x = 5\), \(180^{\circ} < x < 270^{\circ}\).
43 Skill - Inverse Trig · Level 3
Write the given expression as an algebraic expression in \(x\). \(\sin(2 \tan^{-1} x)\).
44 Skill - Inverse Trig · Level 3
Write the given expression as an algebraic expression in \(x\). \(\tan(2 \cos^{-1} x)\).
45 Skill - Inverse Trig · Level 3
Write the given expression as an algebraic expression in \(x\). \(\sin\left(\dfrac{1}{2} \cos^{-1} x\right)\).
46 Skill - Inverse Trig · Level 3
Write the given expression as an algebraic expression in \(x\). \(\cos(2 \sin^{-1} x)\).
47 Skill - Inverse Trig Exact · Level 3
Find the exact value of the given expression. \(\sin(2 \cos^{-1}\left(\dfrac{7}{25}\right))\).
48 Skill - Inverse Trig Exact · Level 3
Find the exact value of the given expression. \(\cos(2 \tan^{-1}\left(\dfrac{12}{5}\right))\).
49 Skill - Inverse Trig Exact · Level 3
Find the exact value of the given expression. \(\sec(2 \sin^{-1}\left(\dfrac{1}{4}\right))\).
50 Skill - Inverse Trig Exact · Level 3
Find the exact value of the given expression. \(\tan(\dfrac{1}{2} \cos^{-1}\left(\dfrac{2}{3}\right))\).
51 Skill - Evaluate · Level 2
Evaluate the expression under the given conditions. \(\cos 2 \theta\); \(\sin \theta = -\dfrac{3}{5}\), \(\theta\) in Quadrant III.
52 Skill - Evaluate · Level 2
Evaluate the expression under the given conditions. \(\sin\left(\dfrac{\theta}{2}\right)\); \(\tan \theta = -\dfrac{5}{12}\), \(\theta\) in Quadrant IV.
53 Skill - Evaluate · Level 2
Evaluate the expression under the given conditions. \(\sin 2 \theta\); \(\sin \theta = \dfrac{1}{7}\), \(\theta\) in Quadrant II.
54 Skill - Evaluate · Level 2
Evaluate the expression under the given conditions. \(\tan 2 \theta\); \(\cos \theta = \dfrac{3}{5}\), \(\theta\) in Quadrant I.
55 Skill - Product to Sum · Level 2
Write the product as a sum. \(\sin 2x \cos 3x\).
56 Skill - Product to Sum · Level 2
Write the product as a sum. \(\sin x \sin 5x\).
57 Skill - Product to Sum · Level 2
Write the product as a sum. \(\cos x \sin 4x\).
58 Skill - Product to Sum · Level 2
Write the product as a sum. \(\cos 5x \cos 3x\).
59 Skill - Product to Sum · Level 2
Write the product as a sum. \(3 \cos 4x \cos 7x\).
60 Skill - Product to Sum · Level 2
Write the product as a sum. \(11 \sin\left(\dfrac{x}{2}\right) \cos\left(\dfrac{x}{4}\right)\).
61 Skill - Sum to Product · Level 2
Write the sum as a product. \(\sin 5x + \sin 3x\).
62 Skill - Sum to Product · Level 2
Write the sum as a product. \(\sin x - \sin 4x\).
63 Skill - Sum to Product · Level 2
Write the sum as a product. \(\cos 4x - \cos 6x\).
64 Skill - Sum to Product · Level 2
Write the sum as a product. \(\cos 9x + \cos 2x\).
65 Skill - Sum to Product · Level 2
Write the sum as a product. \(\sin 2x - \sin 7x\).
66 Skill - Sum to Product · Level 2
Write the sum as a product. \(\sin 3x + \sin 4x\).
67 Skill - Compute Value · Level 2
Find the value of the product or sum. \(2 \sin 52.5^{\circ} \sin 97.5^{\circ}\).
68 Skill - Compute Value · Level 2
Find the value of the product or sum. \(3 \cos 37.5^{\circ} \sin 7.5^{\circ}\).
69 Skill - Compute Value · Level 2
Find the value of the product or sum. \(\cos 37.5^{\circ} \cos 7.5^{\circ}\).
70 Skill - Compute Value · Level 2
Find the value of the product or sum. \(\sin 75^{\circ} + \sin 15^{\circ}\).
71 Skill - Compute Value · Level 2
Find the value of the product or sum. \(\cos 255^{\circ} - \cos 195^{\circ}\).
72 Skill - Compute Value · Level 2
Find the value of the product or sum. \(\cos\left(\dfrac{\pi}{12}\right) + \cos\left(5 \dfrac{\pi}{12}\right)\).
73 Proof - Identity · Level 3
Prove the identity. \(\cos^2 5x - \sin^2 5x = \cos 10x\).
74 Proof - Identity · Level 3
Prove the identity. \(\sin 8x = 2 \sin 4x \cos 4x\).
75 Proof - Identity · Level 3
Prove the identity. \((\sin x + \cos x)^2 = 1 + \sin 2x\).
76 Proof - Identity · Level 3
Prove the identity. \(\dfrac{2 \tan x}{1 + \tan^2 x} = \sin 2x\).
77 Proof - Identity · Level 3
Prove the identity. \(\dfrac{\sin 4x}{\sin x} = 4 \cos x \cos 2x\).
78 Proof - Identity · Level 3
Prove the identity. \(\dfrac{1 + \sin 2x}{\sin 2x} = 1 + \dfrac{1}{2} \sec x \csc x\).
79 Proof - Identity · Level 4
Prove the identity. \(\dfrac{2(\tan x - \cot x)}{\tan^2 x - \cot^2 x} = \sin 2x\).
80 Proof - Identity · Level 3
Prove the identity. \(\cot 2x = \dfrac{1 - \tan^2 x}{2 \tan x}\).
81 Proof - Identity · Level 4
Prove the identity. \(\tan 3x = \dfrac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x}\).
82 Proof - Identity · Level 4
Prove the identity. \(4(\sin^6 x + \cos^6 x) = 4 - 3 \sin^2 2x\).
83 Proof - Identity · Level 3
Prove the identity. \(\cos^4 x - \sin^4 x = \cos 2x\).
84 Proof - Identity · Level 4
Prove the identity. \(\tan^2\left(\dfrac{x}{2} + \dfrac{\pi}{4}\right) = \dfrac{1 + \sin x}{1 - \sin x}\).
85 Proof - Identity · Level 3
Prove the identity. \(\dfrac{\sin x + \sin 5x}{\cos x + \cos 5x} = \tan 3x\).
86 Proof - Identity · Level 3
Prove the identity. \(\dfrac{\sin 3x + \sin 7x}{\cos 3x - \cos 7x} = \cot 2x\).
87 Proof - Identity · Level 3
Prove the identity. \(\dfrac{\sin 10x}{\sin 9x + \sin x} = \dfrac{\cos 5x}{\cos 4x}\).
88 Proof - Identity · Level 4
Prove the identity. \(\dfrac{\sin x + \sin 3x + \sin 5x}{\cos x + \cos 3x + \cos 5x} = \tan 3x\).
89 Proof - Identity · Level 3
Prove the identity. \(\dfrac{\sin x + \sin y}{\cos x + \cos y} = \tan\left(\dfrac{x + y}{2}\right)\).
90 Proof - Identity · Level 4
Prove the identity. \(\tan y = \dfrac{\sin(x + y) - \sin(x - y)}{\cos(x + y) + \cos(x - y)}\).
91 Show · Level 3
Show that \(\sin 130^{\circ} - \sin 110^{\circ} = -\sin 10^{\circ}\).
92 Show · Level 3
Show that \(\cos 100^{\circ} - \cos 200^{\circ} = \sin 50^{\circ}\).
93 Show · Level 3
Show that \(\sin 45^{\circ} + \sin 15^{\circ} = \sin 75^{\circ}\).
94 Show · Level 3
Show that \(\cos 87^{\circ} + \cos 33^{\circ} = \sin 63^{\circ}\).
95 Proof - Identity · Level 4
Prove the identity \(\dfrac{\sin x + \sin 2x + \sin 3x + \sin 4x + \sin 5x}{\cos x + \cos 2x + \cos 3x + \cos 4x + \cos 5x} = \tan 3x\).
96 Proof - Recursive · Level 4
Use the identity \(\sin 2x = 2 \sin x \cos x\) \(n\) times to show that \(\sin(2^n x) = 2^n \sin x \cos x \cos 2x \cos 4x \cdots.c \cos 2^{n-1} x\).
97 Graphing - Conjecture · Level 3
(a) Graph \(f(x) = \dfrac{\sin 3x}{\sin x} - \dfrac{\cos 3x}{\cos x}\) and make a conjecture. (b) Prove the conjecture you made in part (a).
98 Graphing - Conjecture · Level 2
(a) Graph \(f(x) = \cos 2x + 2 \sin^2 x\) and make a conjecture. (b) Prove the conjecture you made in part (a).
99 Graphing · Level 3
Let \(f(x) = \sin 6x + \sin 7x\). (a) Graph \(y = f(x)\). (b) Verify that \(f(x) = 2 \cos\left(\dfrac{1}{2} x\right) \sin\left(\dfrac{13}{2} x\right)\). (c) Graph \(y = 2 \cos\left(\dfrac{1}{2} x\right)\) and \(y = -2 \cos\left(\dfrac{1}{2} x\right)\), together with the graph in part (a), in the same viewing rectangle. How are these graphs related to the graph of \(f\)?
100 Application - Cubic · Level 4
Let \(3x = \dfrac{\pi}{3}\) and let \(y = \cos x\). Use the result of Example 2 to show that \(y\) satisfies the equation \(8 y^3 - 6 y - 1 = 0\).
101 Theory - Tchebycheff · Level 4
(a) Express \(\cos 4x\) as a polynomial of degree 4 in \(\cos x\). (b) Show that there is a polynomial \(Q(t)\) of degree 5 such that \(\cos 5x = Q(\cos x)\).
102 Application - Geometry · Level 4
In triangle \(ABC\) (see the figure) the line segment \(s\) bisects angle \(C\). Show that the length of \(s\) is given by \(s = \dfrac{2 a b \cos x}{a + b}\). [Hint: Use the Law of Sines.]
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103 Application - Triangle · Level 4
If \(A\), \(B\), and \(C\) are the angles in a triangle, show that \(\sin 2A + \sin 2B + \sin 2C = 4 \sin A \sin B \sin C\).
104 Application - Optimization · Level 3
A rectangle is to be inscribed in a semicircle of radius 5 cm as shown in the figure. (a) Show that the area of the rectangle is modeled by the function \(A(\theta) = 25 \sin 2 \theta\). (b) Find the largest possible area for such an inscribed rectangle. (c) Find the dimensions of the inscribed rectangle with the largest possible area.
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105 Application - Wooden Beam · Level 4
Sawing a Wooden Beam. A rectangular beam is to be cut from a cylindrical log of diameter 20 in. (a) Show that the cross-sectional area of the beam is modeled by the function \(A(\theta) = 200 \sin 2 \theta\), where \(\theta\) is as shown in the figure. (b) Show that the maximum cross-sectional area of such a beam is 200 in\(^2\). [Hint: Use the fact that \(\sin u\) achieves its maximum value at \(u = \dfrac{\pi}{2}\).]
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106 Application - Fold · Level 4
Length of a Fold. The lower right-hand corner of a long piece of paper 6 in. wide is folded over to the left-hand edge as shown. The length \(L\) of the fold depends on the angle \(\theta\). Show that \(L = \dfrac{3}{\sin \theta \cos^2 \theta}\).
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107 Application - Sound Beats · Level 3
Sound Beats. When two pure notes that are close in frequency are played together, their sounds interfere to produce beats; that is, the loudness (or amplitude) of the sound alternately increases and decreases. If the two notes are given by \(f_1(t) = \cos 11 t\) and \(f_2(t) = \cos 13 t\), the resulting sound is \(f(t) = f_1(t) + f_2(t)\). (a) Graph the function \(y = f(t)\). (b) Verify that \(f(t) = 2 \cos t \cos 12 t\).
108 Application - Touch-Tone Telephones · Level 4
Touch-Tone Telephones. When a key is pressed on a touch-tone telephone, the keypad generates two pure tones, which combine to produce a sound that uniquely identifies the key. The figure shows the low frequency \(f_1\) and the high frequency \(f_2\) associated with each key. Pressing a key produces the sound wave \(y = \sin(2 \pi f_1 t) + \sin(2 \pi f_2 t)\).
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(a) Find the function that models the sound produced when the 4 key is pressed.
(b) Use a Sum-to-Product Formula to express the sound generated by the 4 key as a product of a sine and a cosine.
(c) Graph the sound wave generated by the 4 key, from \(t = 0\) to \(t = 0.006\) s.

Enter your answer directly below each part above.

109 Discovery - Geometric Proof of Double-Angle Formula · Level 4
Geometric Proof of a Double-Angle Formula. Use the figure to prove that \(\sin 2 \theta = 2 \sin \theta \cos \theta\). [Hint: Find the area of triangle \(\text{ABC}\) in two different ways. You will need the following facts from geometry: An angle inscribed in a semicircle is a right angle, so \(\angle \text{ACB}\) is a right angle. The central angle subtended by the chord of a circle is twice the angle subtended by the chord on the circle, so \(\angle \text{BOC}\) is \(2 \theta\).]
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110 Example - Using the Double-Angle Formulas · Level 2
If \(\cos x = -\dfrac{2}{3}\) and \(x\) is in Quadrant II, find \(\cos 2x\) and \(\sin 2x\).
111 Example - Triple-Angle Formula · Level 3
Write \(\cos 3x\) in terms of \(\cos x\).
112 Example - Proving an Identity · Level 3
Prove the identity \(\dfrac{\sin 3x}{\sin x \cos x} = 4 \cos x - \sec x\).
113 Example - Lowering Powers · Level 3
Express \(\sin^2 x \cos^2 x\) in terms of the first power of cosine.
114 Example - Half-Angle Formula · Level 2
Find the exact value of \(\sin 22.5^{\circ}\).
115 Example - Half-Angle Formula · Level 3
Find \(\tan\left(\dfrac{u}{2}\right)\) if \(\sin u = \dfrac{2}{5}\) and \(u\) is in Quadrant II.
116 Example - Inverse Trig Functions · Level 3
Write \(\sin(2 \cos^{-1} x)\) as an algebraic expression in \(x\) only, where \(-1 \leq x \leq 1\).
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117 Example - Inverse Trig Functions · Level 3
Evaluate \(\sin 2 \theta\), where \(\cos \theta = -\dfrac{2}{5}\) with \(\theta\) in Quadrant II.
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118 Example - Product to Sum · Level 2
Express \(\sin 3x \sin 5x\) as a sum of trigonometric functions.
119 Example - Sum to Product · Level 2
Write \(\sin 7x + \sin 3x\) as a product.
120 Example - Proving an Identity · Level 3
Verify the identity \(\dfrac{\sin 3x - \sin x}{\cos 3x + \cos x} = \tan x\).

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