Textbook Question
Verify that each equation is an identity.
(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1
Ch. 5 - Trigonometric Identities
Lial12th EditionTrigonometryISBN: 9780136552161
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 6, Problem 5.62
Verify that each equation is an identity.
1/(sec α - tan α) = sec α + tan α
Verified step by step guidance1
Start by expressing \( \sec \alpha \) and \( \tan \alpha \) in terms of sine and cosine: \( \sec \alpha = \frac{1}{\cos \alpha} \) and \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \).
Substitute these expressions into the left-hand side of the equation: \( \frac{1}{\sec \alpha - \tan \alpha} = \frac{1}{\frac{1}{\cos \alpha} - \frac{\sin \alpha}{\cos \alpha}} \).
Simplify the denominator by combining the fractions: \( \frac{1}{\frac{1 - \sin \alpha}{\cos \alpha}} = \frac{\cos \alpha}{1 - \sin \alpha} \).
Now, consider the right-hand side: \( \sec \alpha + \tan \alpha = \frac{1}{\cos \alpha} + \frac{\sin \alpha}{\cos \alpha} = \frac{1 + \sin \alpha}{\cos \alpha} \).
Multiply the numerator and the denominator of the left-hand side by the conjugate of the denominator \( 1 + \sin \alpha \) to simplify and verify that both sides are equal: \( \frac{\cos \alpha (1 + \sin \alpha)}{(1 - \sin \alpha)(1 + \sin \alpha)} = \frac{1 + \sin \alpha}{\cos \alpha} \).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for verifying equations and simplifying expressions in trigonometry.
Recommended video:
5:32
Fundamental Trigonometric Identities
Reciprocal Functions
Reciprocal functions in trigonometry include secant (sec) and cosecant (csc), which are the reciprocals of cosine and sine, respectively. The secant function is defined as sec α = 1/cos α, and the tangent function is defined as tan α = sin α/cos α. Recognizing these relationships is essential for manipulating and verifying trigonometric equations.
Recommended video:
3:23Secant, Cosecant, & Cotangent on the Unit Circle
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This includes factoring, expanding, and combining like terms. In the context of trigonometric identities, effective algebraic manipulation allows one to transform one side of an equation to match the other, thereby verifying the identity.
Recommended video:
04:12Algebraic Operations on Vectors
Related Practice
Textbook Question
Identify the basic trigonometric function graphed, and determine whether it is even or odd.
<IMAGE>
Textbook Question
Verify that each equation is an identity.
(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ
Textbook Question
Factor each trigonometric expression.
cot⁴ x + 3 cot² x + 2
Textbook Question
Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
cos θ (cos θ - sec θ)
Textbook Question
Simplify each expression.
± √[(1 + cos (x/4))/2]