Problem 57
Question
Verify the identity by transforming the lefthand side into the right-hand side. $$\cos ^{2} \theta\left(\sec ^{2} \theta-1\right)=\sin ^{2} \theta$$
Step-by-Step Solution
Verified Answer
The identity is verified as both sides equal \( \sin^2 \theta \).
1Step 1: Recognize the Identity Involving Secant and Cosine
Begin by recalling the identity for secant: \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \).
2Step 2: Substitute and Simplify the Left Side
Starting with the left-hand side, \( \cos^2 \theta (\sec^2 \theta - 1) \), substitute \( \sec^2 \theta \) with \( \frac{1}{\cos^2 \theta} \). This gives: \[ \cos^2 \theta \left( \frac{1}{\cos^2 \theta} - 1 \right) \].
3Step 3: Simplify the Expression Inside the Parentheses
Simplify the expression inside the parentheses: \( \frac{1}{\cos^2 \theta} - 1 = \frac{1 - \cos^2 \theta}{\cos^2 \theta} \).
4Step 4: Multiply and Simplify Further
Multiply \( \cos^2 \theta \) by \( \frac{1 - \cos^2 \theta}{\cos^2 \theta} \), which simplifies to \( 1 - \cos^2 \theta \).
5Step 5: Recognize the Pythagorean Identity
Using the Pythagorean identity \( \sin^2 \theta = 1 - \cos^2 \theta \), recognize that the expression \( 1 - \cos^2 \theta \) is equivalent to \( \sin^2 \theta \).
6Step 6: Conclude with Verification
Thus, the left-hand side \( \cos^2 \theta (\sec^2 \theta - 1) \) simplifies to \( \sin^2 \theta \), verifying the given identity.
Key Concepts
Pythagorean identityTrigonometric functionsAlgebraic manipulation
Pythagorean identity
The Pythagorean identity is one of the most fundamental identities in trigonometry. It states that for any angle \( \theta \), the square of the sine function plus the square of the cosine function equals 1. Mathematically, this is expressed as:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity underpins many other identities in trigonometry and is derived from the Pythagorean theorem, which relates the lengths of the sides of a right triangle. In the solution above, we used a variation of this identity to transform the expression into something recognizable as \( \sin^2 \theta \). This identity is crucial when manipulating trigonometric expressions because it allows us to substitute one function for another, simplifying complex equations.
Whenever you see an expression like \( 1 - \cos^2 \theta \), you should immediately recognize it as \( \sin^2 \theta \), based on rearranging the Pythagorean identity. This insight can simplify many trigonometric proofs and problems, as shown in the step where the solution recognizes \( 1 - \cos^2 \theta \) as \( \sin^2 \theta \).
Another common form of the Pythagorean identity is expressing \( \cos^2 \theta \) in terms of \( \sin^2 \theta \):\[\cos^2 \theta = 1 - \sin^2 \theta\]These variations are helpful in a wide range of problems, so it's vital to become familiar with these identities for efficient problem-solving.
Whenever you see an expression like \( 1 - \cos^2 \theta \), you should immediately recognize it as \( \sin^2 \theta \), based on rearranging the Pythagorean identity. This insight can simplify many trigonometric proofs and problems, as shown in the step where the solution recognizes \( 1 - \cos^2 \theta \) as \( \sin^2 \theta \).
Another common form of the Pythagorean identity is expressing \( \cos^2 \theta \) in terms of \( \sin^2 \theta \):\[\cos^2 \theta = 1 - \sin^2 \theta\]These variations are helpful in a wide range of problems, so it's vital to become familiar with these identities for efficient problem-solving.
Trigonometric functions
Trigonometric functions are essential tools in mathematics that relate the angles of a triangle to the ratios of its sides. In trigonometry, common functions include sine (\( \sin \)), cosine (\( \cos \)), and secant (\( \sec \)), among others. Each function has its unique properties and identities, which allow us to analyze and solve various geometric and mathematical problems.
The secant function is defined as the reciprocal of the cosine function:\[\sec \theta = \frac{1}{\cos \theta}\]Thus, \( \sec^2 \theta = \left( \frac{1}{\cos \theta} \right)^2 = \frac{1}{\cos^2 \theta} \), as used in the solution above. This definition confirms that if you know the value of \( \cos \theta \), you can easily find \( \sec \theta \). Recognizing these relationships is key to simplifying and solving trigonometric expressions.
Understanding how these functions interrelate through identities and equations is vital. It allows for the transformation and simplification of expressions, as seen when substituting secant with cosine, which simplifies the process of verifying identities or solving equations. Mastering these core functions lays the groundwork for more advanced studies in trigonometry and calculus.
The secant function is defined as the reciprocal of the cosine function:\[\sec \theta = \frac{1}{\cos \theta}\]Thus, \( \sec^2 \theta = \left( \frac{1}{\cos \theta} \right)^2 = \frac{1}{\cos^2 \theta} \), as used in the solution above. This definition confirms that if you know the value of \( \cos \theta \), you can easily find \( \sec \theta \). Recognizing these relationships is key to simplifying and solving trigonometric expressions.
Understanding how these functions interrelate through identities and equations is vital. It allows for the transformation and simplification of expressions, as seen when substituting secant with cosine, which simplifies the process of verifying identities or solving equations. Mastering these core functions lays the groundwork for more advanced studies in trigonometry and calculus.
Algebraic manipulation
Algebraic manipulation is the method of rearranging and simplifying mathematical expressions using algebraic rules and identities. In the context of trigonometric identities, it often involves substituting known identities and simplifying expressions to reveal or prove an equivalency.
For example, in solving the given expression, the process begins by substituting \( \sec^2 \theta \) with \( \frac{1}{\cos^2 \theta} \). This substitution is a direct application of understanding the reciprocal identity of secant. The next step involves simplifying the expression inside the parentheses:\[\frac{1}{\cos^2 \theta} - 1 = \frac{1 - \cos^2 \theta}{\cos^2 \theta}\]This requires the careful handling of fractions and common denominators, skills that are crucial in algebraic manipulation.
Following simplification, multiplying and reducing further leads to recognizing a familiar pattern or identity. Identifying \( 1 - \cos^2 \theta \) as \( \sin^2 \theta \) is an example of using a trigonometric identity to simplify an expression. This step is a key part of many algebraic manipulations in trigonometry, as it allows for transitioning between different trigonometric functions.
By practicing algebraic manipulation, you gain a deeper understanding of how mathematical rules interact and how to effectively apply them to reach a solution. This skill is invaluable in mathematics, especially in solving complex trigonometric and calculus problems.
For example, in solving the given expression, the process begins by substituting \( \sec^2 \theta \) with \( \frac{1}{\cos^2 \theta} \). This substitution is a direct application of understanding the reciprocal identity of secant. The next step involves simplifying the expression inside the parentheses:\[\frac{1}{\cos^2 \theta} - 1 = \frac{1 - \cos^2 \theta}{\cos^2 \theta}\]This requires the careful handling of fractions and common denominators, skills that are crucial in algebraic manipulation.
Following simplification, multiplying and reducing further leads to recognizing a familiar pattern or identity. Identifying \( 1 - \cos^2 \theta \) as \( \sin^2 \theta \) is an example of using a trigonometric identity to simplify an expression. This step is a key part of many algebraic manipulations in trigonometry, as it allows for transitioning between different trigonometric functions.
By practicing algebraic manipulation, you gain a deeper understanding of how mathematical rules interact and how to effectively apply them to reach a solution. This skill is invaluable in mathematics, especially in solving complex trigonometric and calculus problems.
Other exercises in this chapter
Problem 57
Scientists sometimes use the formula \(f(t)=a \sin (b t+c)+d\) to simulate temperature variations during the day, with time \(t\) in hours, temperature \(f(t)\)
View solution Problem 57
Use the graph of a trigonometric function to ald in sketching the graph of the equation without plotting points. $$y=|\sin x|+2$$
View solution Problem 58
Use the graph of a trigonometric function to ald in sketching the graph of the equation without plotting points. $$y=|\cos x|-3$$
View solution Problem 58
Verify the identity by transforming the lefthand side into the right-hand side. $$(\tan \theta+\cot \theta) \tan \theta=\sec ^{2} \theta$$
View solution