Problem 52
Question
Verify each identity. \(\cos 2 \alpha=\cos ^{2} \alpha-\sin ^{2} \alpha\) Hint: Write \(\cos 2 \alpha\) as \(\cos (\alpha+\alpha)\)
Step-by-Step Solution
Verified Answer
Both LHS and RHS are equal to \(\cos^{2} \alpha - \sin^{2} \alpha\). Hence, the identity is verified.
1Step 1: Rewrite Using Cosine of Sum of Angles Identity
Rewrite \(\cos 2 \alpha\) as \(\cos (\alpha+\alpha)\). According to the Cosine of Sum of Angles Identity, \( \cos (A+B)=\cos A \cos B - \sin A \sin B \). So we can write \(\cos 2 \alpha\) as \(\cos \alpha \cos \alpha - \sin \alpha \sin \alpha\) or \(\cos ^{2} \alpha - \sin ^{2} \alpha\).
2Step 2: Prove that \(\cos ^{2} \alpha - \sin ^{2} \alpha = \cos 2 \alpha\)
The Right-hand side (RHS) is already in the form of \(\cos ^{2} \alpha - \sin ^{2} \alpha\). The Left-hand side (LHS) \(\cos 2 \alpha\) was rewritten as \(\cos ^{2} \alpha - \sin ^{2} \alpha\) in step 1. Since both LHS and RHS are equal, the identity is verified.
Key Concepts
Trigonometric IdentitiesVerifying Trigonometric IdentitiesDouble Angle FormulasCosine Squared and Sine Squared Relationship
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. These identities are essential tools for simplifying expressions and solving trigonometry problems, acting as the backbone of the subject.
One of the fundamental reasons we use identities in trigonometry is to transform complex expressions into simpler ones so that we can solve equations, integrate, differentiate, and perform other mathematical operations more easily. Common trigonometric identities include reciprocal identities, quotient identities, Pythagorean identities, and angle sum and difference identities. These are derived from the unit circle and the relationships between the sides and angles of a right triangle.
One of the fundamental reasons we use identities in trigonometry is to transform complex expressions into simpler ones so that we can solve equations, integrate, differentiate, and perform other mathematical operations more easily. Common trigonometric identities include reciprocal identities, quotient identities, Pythagorean identities, and angle sum and difference identities. These are derived from the unit circle and the relationships between the sides and angles of a right triangle.
Verifying Trigonometric Identities
To verify a trigonometric identity means to prove that the equation is true for all values within the domain of the variables involved. Verifying an identity often requires a succession of algebraic steps where you manipulate one or both sides of the equation using known identities until you arrive at a point where both sides are identical or can be directly compared.
When verifying trigonometric identities, a good strategy is to start with the more complex side and transform it into the simpler side. In some cases, you may need to work on both sides of the equation. Algebraic techniques like factoring, expanding, and consolidating fractions are often employed. Additionally, strategic substitution based on known identities is a crucial tool. Patience and practice are key, as verification problems can become quite intricate.
When verifying trigonometric identities, a good strategy is to start with the more complex side and transform it into the simpler side. In some cases, you may need to work on both sides of the equation. Algebraic techniques like factoring, expanding, and consolidating fractions are often employed. Additionally, strategic substitution based on known identities is a crucial tool. Patience and practice are key, as verification problems can become quite intricate.
Double Angle Formulas
Double angle formulas are specific trigonometric identities that express trigonometric functions of double angles — for instance, \(2\alpha\) — in terms of single angles like \(\alpha\). These are particularly useful in calculus for integrating powers of trigonometric functions and in solving trigonometric equations.
For the cosine function, the double angle formula is: \(\cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha)\). This identity is derivable from the more general angle sum formula for cosine, where both angles are equal (i.e., \(\alpha = \beta\)). Other double angle formulas include those for the sine and tangent functions, and each can be further transformed using Pythagorean identities to offer alternative forms, useful in various contexts.
For the cosine function, the double angle formula is: \(\cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha)\). This identity is derivable from the more general angle sum formula for cosine, where both angles are equal (i.e., \(\alpha = \beta\)). Other double angle formulas include those for the sine and tangent functions, and each can be further transformed using Pythagorean identities to offer alternative forms, useful in various contexts.
Cosine Squared and Sine Squared Relationship
The relationship between the square of the cosine function and the square of the sine function is a part of the fundamental Pythagorean identity in trigonometry. This relation states that for any angle \(\alpha\), the sum of the squares of cosine and sine of \(\alpha\) is equal to 1: \(\cos^2(\alpha) + \sin^2(\alpha) = 1\).
This identity is derived from the Pythagorean theorem applied to a unit circle. Its importance lies in the ability to convert expressions containing \(\sin^2(\alpha)\) to expressions with \(\cos^2(\alpha)\) and vice versa, greatly simplifying the process of verifying identities or solving equations. For example, to find \(\cos^2(\alpha)\), one may subtract \(\sin^2(\alpha)\) from one, and similarly to find \(\sin^2(\alpha)\), one may subtract \(\cos^2(\alpha)\) from one.
This identity is derived from the Pythagorean theorem applied to a unit circle. Its importance lies in the ability to convert expressions containing \(\sin^2(\alpha)\) to expressions with \(\cos^2(\alpha)\) and vice versa, greatly simplifying the process of verifying identities or solving equations. For example, to find \(\cos^2(\alpha)\), one may subtract \(\sin^2(\alpha)\) from one, and similarly to find \(\sin^2(\alpha)\), one may subtract \(\cos^2(\alpha)\) from one.
Other exercises in this chapter
Problem 52
Verify each identity. $$\sin ^{4} t-\cos ^{4} t=1-2 \cos ^{2} t$$
View solution Problem 52
Involve trigonometric equations quadratic in form. Solve each equation on the interval \([0,2 \pi)\) $$4 \sec ^{2} x-2=0$$
View solution Problem 53
Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can use the sum and difference formulas for cosines and sines
View solution Problem 53
Verify each identity. $$\frac{\sin \theta-\cos \theta}{\sin \theta}+\frac{\cos \theta-\sin \theta}{\cos \theta}=2-\sec \theta \csc \theta$$
View solution