Stewart Precalc 6e Section 7.1: Trigonometric Identities

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Stewart Precalc 6e Section 7.1: Trigonometric Identities 0/109
1 Concept - Identity Definition · Level 1
An equation is called an identity if it is valid for ____ values of the variable. The equation \(2x = x + x\) is an algebraic identity, and the equation \(\sin^2 x + \cos^2 x = \)____ is a trigonometric identity.
2 Concept - Even-Odd Identity · Level 1
For any \(x\) it is true that \(\cos(-x)\) has the same value as \(\cos x\). We express this fact as the identity ____.
3 Skills - Simplify via Sine/Cosine · Level 2
Write the trigonometric expression in terms of sine and cosine, and then simplify: \(\cos t \tan t\).
4 Skills - Simplify via Sine/Cosine · Level 2
Write the trigonometric expression in terms of sine and cosine, and then simplify: \(\cos t \csc t\).
5 Skills - Simplify via Sine/Cosine · Level 2
Write the trigonometric expression in terms of sine and cosine, and then simplify: \(\sin \theta \sec \theta\).
6 Skills - Simplify via Sine/Cosine · Level 2
Write the trigonometric expression in terms of sine and cosine, and then simplify: \(\tan \theta \csc \theta\).
7 Skills - Simplify via Sine/Cosine · Level 2
Write the trigonometric expression in terms of sine and cosine, and then simplify: \(\tan^2 x - \sec^2 x\).
8 Skills - Simplify via Sine/Cosine · Level 2
Write the trigonometric expression in terms of sine and cosine, and then simplify: \(\dfrac{\sec x}{\csc x}\).
9 Skills - Simplify via Sine/Cosine · Level 2
Write the trigonometric expression in terms of sine and cosine, and then simplify: \(\sin u + \cot u \cos u\).
10 Skills - Simplify via Sine/Cosine · Level 2
Write the trigonometric expression in terms of sine and cosine, and then simplify: \(\cos^2 \theta (1 + \tan^2 \theta)\).
11 Skills - Simplify via Sine/Cosine · Level 2
Write the trigonometric expression in terms of sine and cosine, and then simplify: \(\dfrac{\sec \theta - \cos \theta}{\sin \theta}\).
12 Skills - Simplify via Sine/Cosine · Level 2
Write the trigonometric expression in terms of sine and cosine, and then simplify: \(\dfrac{\cot \theta}{\csc \theta - \sin \theta}\).
13 Skills - Simplify Expression · Level 2
Simplify the trigonometric expression: \(\dfrac{\sin x \sec x}{\tan x}\).
14 Skills - Simplify Expression · Level 2
Simplify the trigonometric expression: \(\cos^3 x + \sin^2 x \cos x\).
15 Skills - Simplify Expression · Level 2
Simplify the trigonometric expression: \(\dfrac{1 + \cos y}{1 + \sec y}\).
16 Skills - Simplify Expression · Level 2
Simplify the trigonometric expression: \(\dfrac{\tan x}{\sec(-x)}\).
17 Skills - Simplify Expression · Level 2
Simplify the trigonometric expression: \(\dfrac{\sec^2 x - 1}{\sec^2 x}\).
18 Skills - Simplify Expression · Level 2
Simplify the trigonometric expression: \(\dfrac{\sec x - \cos x}{\tan x}\).
19 Skills - Simplify Expression · Level 3
Simplify the trigonometric expression: \(\dfrac{1 + \csc x}{\cos x + \cot x}\).
20 Skills - Simplify Expression · Level 2
Simplify the trigonometric expression: \(\dfrac{\sin x}{\csc x} + \dfrac{\cos x}{\sec x}\).
21 Skills - Simplify Expression · Level 3
Simplify the trigonometric expression: \(\dfrac{1 + \sin u}{\cos u} + \dfrac{\cos u}{1 + \sin u}\).
22 Skills - Simplify Expression · Level 2
Simplify the trigonometric expression: \(\tan x \cos x \csc x\).
23 Skills - Simplify Expression · Level 3
Simplify the trigonometric expression: \(\dfrac{2 + \tan^2 x}{\sec^2 x} - 1\).
24 Skills - Simplify Expression · Level 2
Simplify the trigonometric expression: \(\dfrac{1 + \cot A}{\csc A}\).
25 Skills - Simplify Expression · Level 2
Simplify the trigonometric expression: \(\tan \theta + \cos(-\theta) + \tan(-\theta)\).
26 Skills - Simplify Expression · Level 3
Simplify the trigonometric expression: \(\dfrac{\cos x}{\sec x + \tan x}\).
27 Skills - Verify Identity Algebraically and Graphically · Level 2
Consider the equation \(\dfrac{\cos x}{\sec x \sin x} = \csc x - \sin x\). (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity.
28 Skills - Verify Identity Algebraically and Graphically · Level 2
Consider the equation \(\dfrac{\tan y}{\csc y} = \sec y - \cos y\). (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity.
29 Skills - Verify Identity · Level 2
Verify the identity \(\dfrac{\sin \theta}{\tan \theta} = \cos \theta\).
30 Skills - Verify Identity · Level 2
Verify the identity \(\dfrac{\tan x}{\sec x} = \sin x\).
31 Skills - Verify Identity · Level 2
Verify the identity \(\dfrac{\cos u \sec u}{\tan u} = \cot u\).
32 Skills - Verify Identity · Level 2
Verify the identity \(\dfrac{\cot x \sec x}{\csc x} = 1\).
33 Skills - Verify Identity · Level 2
Verify the identity \(\sin B + \cos B \cot B = \csc B\).
34 Skills - Verify Identity · Level 2
Verify the identity \(\cos(-x) - \sin(-x) = \cos x + \sin x\).
35 Skills - Verify Identity · Level 2
Verify the identity \(\cot(-\alpha) \cos(-\alpha) + \sin(-\alpha) = -\csc \alpha\).
36 Skills - Verify Identity · Level 2
Verify the identity \(\csc x [\csc x + \sin(-x)] = \cot^2 x\).
37 Skills - Verify Identity · Level 2
Verify the identity \(\tan \theta + \cot \theta = \sec \theta \csc \theta\).
38 Skills - Verify Identity · Level 2
Verify the identity \((\sin x + \cos x)^2 = 1 + 2 \sin x \cos x\).
39 Skills - Verify Identity · Level 2
Verify the identity \((1 - \cos \beta)(1 + \cos \beta) = \dfrac{1}{\csc^2 \beta}\).
40 Skills - Verify Identity · Level 2
Verify the identity \(\dfrac{\cos x}{\sec x} + \dfrac{\sin x}{\csc x} = 1\).
41 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{(\sin x + \cos x)^2}{\sin^2 x - \cos^2 x} = \dfrac{\sin^2 x - \cos^2 x}{(\sin x - \cos x)^2}\).
42 Skills - Verify Identity · Level 2
Verify the identity \((\sin x + \cos x)^4 = (1 + 2 \sin x \cos x)^2\).
43 Skills - Verify Identity · Level 2
Verify the identity \(\dfrac{\sec t - \cos t}{\sec t} = \sin^2 t\).
44 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{1 - \sin x}{1 + \sin x} = (\sec x - \tan x)^2\).
45 Skills - Verify Identity · Level 2
Verify the identity \(\dfrac{1}{1 - \sin^2 y} = 1 + \tan^2 y\).
46 Skills - Verify Identity · Level 2
Verify the identity \(\csc x - \sin x = \cos x \cot x\).
47 Skills - Verify Identity · Level 3
Verify the identity \((\cot x - \csc x)(\cos x + 1) = -\sin x\).
48 Skills - Verify Identity · Level 2
Verify the identity \(\sin^4 \theta - \cos^4 \theta = \sin^2 \theta - \cos^2 \theta\).
49 Skills - Verify Identity · Level 2
Verify the identity \((1 - \cos^2 x)(1 + \cot^2 x) = 1\).
50 Skills - Verify Identity · Level 2
Verify the identity \(2 \cos^2 x - 1 = 1 - 2 \sin^2 x\).
51 Skills - Verify Identity · Level 2
Verify the identity \((\tan y + \cot y) \sin y \cos y = 1\).
52 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{1 - \cos \alpha}{\sin \alpha} = \dfrac{\sin \alpha}{1 + \cos \alpha}\).
53 Skills - Verify Identity · Level 2
Verify the identity \(\sin^2 \alpha + \cos^2 \alpha + \tan^2 \alpha = \sec^2 \alpha\).
54 Skills - Verify Identity · Level 3
Verify the identity \(\tan^2 \theta - \sin^2 \theta = \tan^2 \theta \sin^2 \theta\).
55 Skills - Verify Identity · Level 3
Verify the identity \(\cot^2 \theta \cos^2 \theta = \cot^2 \theta - \cos^2 \theta\).
56 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\sin x - 1}{\sin x + 1} = \dfrac{-\cos^2 x}{(\sin x + 1)^2}\).
57 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\sin w}{\sin w + \cos w} = \dfrac{\tan w}{1 + \tan w}\).
58 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{(\sin t + \cos t)^2}{\sin t \cos t} = 2 + \sec t \csc t\).
59 Skills - Verify Identity · Level 3
Verify the identity \(\sec t \csc t (\tan t + \cot t) = \sec^2 t + \csc^2 t\).
60 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{1 + \tan^2 u}{1 - \tan^2 u} = \dfrac{1}{\cos^2 u - \sin^2 u}\).
61 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{1 + \sec^2 x}{1 + \tan^2 x} = 1 + \cos^2 x\).
62 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\sec x}{\sec x - \tan x} = \sec x (\sec x + \tan x)\).
63 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\sec x + \csc x}{\tan x + \cot x} = \sin x + \cos x\).
64 Skills - Verify Identity · Level 2
Verify the identity \(\sec v - \tan v = \dfrac{1}{\sec v + \tan v}\).
65 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\sin A}{1 - \cos A} - \cot A = \csc A\).
66 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\sin x + \cos x}{\sec x + \csc x} = \sin x \cos x\).
67 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{1 - \cos x}{\sin x} + \dfrac{\sin x}{1 - \cos x} = 2 \csc x\).
68 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\csc x - \cot x}{\sec x - 1} = \cot x\).
69 Skills - Verify Identity · Level 2
Verify the identity \(\dfrac{\csc^2 x - \cot^2 x}{\sec^2 x} = \cos^2 x\).
70 Skills - Verify Identity · Level 3
Verify the identity \(\tan^2 u - \sin^2 u = \tan^2 u \sin^2 u\).
71 Skills - Verify Identity · Level 4
Verify the identity \(\dfrac{\tan v \sin v}{\tan v + \sin v} = \dfrac{\tan v - \sin v}{\tan v \sin v}\).
72 Skills - Verify Identity · Level 3
Verify the identity \(\sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x\).
73 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\cos \theta}{1 - \sin \theta} = \sec \theta + \tan \theta\).
74 Skills - Verify Identity · Level 4
Verify the identity \(\dfrac{\cos \theta}{1 - \sin \theta} = \dfrac{\sin \theta - \csc \theta}{\cos \theta - \cot \theta}\).
75 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{1 + \tan x}{1 - \tan x} = \dfrac{\cos x + \sin x}{\cos x - \sin x}\).
76 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\cos^2 t + \tan^2 t - 1}{\sin^2 t} = \tan^2 t\).
77 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{1}{1 - \sin x} - \dfrac{1}{1 + \sin x} = 2 \sec x \tan x\).
78 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{1}{\sec x + \tan x} + \dfrac{1}{\sec x - \tan x} = 2 \sec x\).
79 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{1 + \sin x}{1 - \sin x} - \dfrac{1 - \sin x}{1 + \sin x} = 4 \tan x \sec x\).
80 Skills - Verify Identity · Level 3
Verify the identity \((\tan x + \cot x)^2 = \sec^2 x + \csc^2 x\).
81 Skills - Verify Identity · Level 2
Verify the identity \(\tan^2 x - \cot^2 x = \sec^2 x - \csc^2 x\).
82 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\sec u - 1}{\sec u + 1} = \dfrac{1 - \cos u}{1 + \cos u}\).
83 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\cot x + 1}{\cot x - 1} = \dfrac{1 + \tan x}{1 - \tan x}\).
84 Skills - Verify Identity · Level 4
Verify the identity \(\dfrac{\sin^3 x + \cos^3 x}{\sin x + \cos x} = 1 - \sin x \cos x\).
85 Skills - Verify Identity · Level 4
Verify the identity \(\dfrac{\tan v - \cot v}{\tan^2 v - \cot^2 v} = \sin v \cos v\).
86 Skills - Verify Identity · Level 4
Verify the identity \(\dfrac{1 + \sin x}{1 - \sin x} = (\tan x + \sec x)^2\).
87 Skills - Verify Identity · Level 3
Verify the identity \(\dfrac{\tan x + \tan y}{\cot x + \cot y} = \tan x \tan y\).
88 Skills - Verify Identity · Level 3
Verify the identity \((\tan x + \cot x)^4 = \csc^4 x \sec^4 x\).
89 Skills - Verify Identity · Level 4
Verify the identity \((\sin \alpha - \tan \alpha)(\cos \alpha - \cot \alpha) = (\cos \alpha - 1)(\sin \alpha - 1)\).
90 Skills - Trigonometric Substitution · Level 3
Make the indicated trigonometric substitution in the given algebraic expression and simplify: \(\dfrac{x}{\sqrt{1 - x^2}}\), \(x = \sin \theta\). Assume that \(0 \leq \theta < \dfrac{\pi}{2}\).
91 Skills - Trigonometric Substitution · Level 3
Make the indicated trigonometric substitution in the given algebraic expression and simplify: \(\sqrt{1 + x^2}\), \(x = \tan \theta\). Assume that \(0 \leq \theta < \dfrac{\pi}{2}\).
92 Skills - Trigonometric Substitution · Level 3
Make the indicated trigonometric substitution in the given algebraic expression and simplify: \(\sqrt{x^2 - 1}\), \(x = \sec \theta\). Assume that \(0 \leq \theta < \dfrac{\pi}{2}\).
93 Skills - Trigonometric Substitution · Level 4
Make the indicated trigonometric substitution in the given algebraic expression and simplify: \(\dfrac{1}{x^2 \sqrt{4 + x^2}}\), \(x = 2 \tan \theta\). Assume that \(0 \leq \theta < \dfrac{\pi}{2}\).
94 Skills - Trigonometric Substitution · Level 3
Make the indicated trigonometric substitution in the given algebraic expression and simplify: \(\sqrt{9 - x^2}\), \(x = 3 \sin \theta\). Assume that \(0 \leq \theta < \dfrac{\pi}{2}\).
95 Skills - Trigonometric Substitution · Level 3
Make the indicated trigonometric substitution in the given algebraic expression and simplify: \(\dfrac{\sqrt{x^2 - 25}}{x}\), \(x = 5 \sec \theta\). Assume that \(0 \leq \theta < \dfrac{\pi}{2}\).
96 Skills - Graph and Verify · Level 3
Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x) = g(x)\) is an identity? Prove your answer. \(f(x) = \cos^2 x - \sin^2 x\), \(g(x) = 1 - 2 \sin^2 x\).
97 Skills - Graph and Verify · Level 3
Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x) = g(x)\) is an identity? Prove your answer. \(f(x) = \tan x (1 + \sin x)\), \(g(x) = \dfrac{\sin x \cos x}{1 + \sin x}\).
98 Skills - Graph and Verify · Level 3
Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x) = g(x)\) is an identity? Prove your answer. \(f(x) = (\sin x + \cos x)^2\), \(g(x) = 1\).
99 Skills - Graph and Verify · Level 3
Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x) = g(x)\) is an identity? Prove your answer. \(f(x) = \cos^4 x - \sin^4 x\), \(g(x) = 2 \cos^2 x - 1\).
100 Skills - Show Not an Identity · Level 2
Show that the equation is not an identity. (a) \(\sin 2x = 2 \sin x\). (b) \(\sin(x + y) = \sin x + \sin y\). (c) \(\sec^2 x + \csc^2 x = 1\). (d) \(\dfrac{1}{\sin x + \cos x} = \csc x + \sec x\).
101 Discovery - Cofunction Identities · Level 3
Cofunction Identities. In the right triangle shown, explain why \(v = \left(\dfrac{\pi}{2}\right) - u\). Explain how you can obtain all six cofunction identities from this triangle for \(0 < u < \dfrac{\pi}{2}\).
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102 Discovery - Make Your Own Identity · Level 3
Making Up Your Own Identity. If you start with a trigonometric expression and rewrite it or simplify it, then setting the original expression equal to the rewritten expression yields a trigonometric identity. For instance, from Example 1 we get the identity \(\cos t + \tan t \sin t = \sec t\). Use this technique to make up your own identity, then give it to a classmate to verify.
103 Example - Simplifying a Trigonometric Expression · Level 2
Simplify the expression \(\cos t + \tan t \sin t\).
104 Example - Simplifying a Trigonometric Expression · Level 2
Simplify the expression \(\dfrac{\sin \theta}{\cos \theta} + \dfrac{\cos \theta}{1 + \sin \theta}\).
105 Example - Proving an Identity by Rewriting in Terms of Sine and Cosine · Level 2
Consider the equation \(\cos \theta (\sec \theta - \cos \theta) = \sin^2 \theta\). (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity.
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106 Example - Proving an Identity by Combining Fractions · Level 3
Verify the identity \(2 \tan x \sec x = \dfrac{1}{1 - \sin x} - \dfrac{1}{1 + \sin x}\).
107 Example - Proving an Identity by Introducing Something Extra · Level 3
Verify the identity \(\dfrac{\cos u}{1 - \sin u} = \sec u + \tan u\).
108 Example - Proving an Identity by Working with Both Sides Separately · Level 3
Verify the identity \(\dfrac{1 + \cos \theta}{\cos \theta} = \dfrac{\tan^2 \theta}{\sec \theta - 1}\).
109 Example - Trigonometric Substitution · Level 3
Substitute \(\sin \theta\) for \(x\) in the expression \(\sqrt{1 - x^2}\) and simplify. Assume that \(0 \leq \theta \leq \dfrac{\pi}{2}\).

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