Problem 52 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \begin{aligned} &\cos 2 \alpha=\cos ^{2} \alpha-\sin ^{2} \alpha\\\ &\text {Hint: Write } \cos 2 \alpha \text { as } \cos (\alpha+\alpha) \end{aligned} $$

Step-by-Step Solution

Answer

The identity \( \cos 2 \alpha = \cos^{2} \alpha - \sin^{2} \alpha \) is indeed correct, as it has been derived using basic trigonometric identities.

Step 1: Apply Cosine for the Sum of Two Angles

We begin by expressing \( \cos 2 \alpha \) as \( \cos (\alpha + \alpha) \). Using the identity for the cosine of a sum, this can be rewritten as \( \cos(\alpha + \alpha) = \cos \alpha \cos \alpha - \sin \alpha \sin \alpha \). Therefore, \( \cos 2 \alpha = \cos^{2} \alpha - \sin^{2} \alpha \).

Step 2: Simplify the Right Side

We know that the square of cosine and sine of the same angle are equal to 1 when summed, \( \cos^{2} \alpha + \sin^{2} \alpha = 1 \). Hence, \( \cos^{2} \alpha - \sin^{2} \alpha \) can be simplified to \( \cos^{2} \alpha - (1 - \cos^{2} \alpha) \), which simplifies to \( 2\cos^{2} \alpha - 1 \). So, \( \cos 2 \alpha = 2\cos^{2} \alpha - 1 \).

Step 3: Conclude the Verification

We've shown that \( \cos 2 \alpha \) can be expressed as \( 2\cos^{2} \alpha - 1 \), which indicates that the provided identity is correct.

Key Concepts

Cosine of a Sum

Understanding the cosine of a sum is crucial when delving into trigonometry. This concept allows us to express the cosine of the sum of two angles in terms of the product of their sines and cosines. Specifically, the identity is given as
\[ \text{cos}(\theta + \rho) = \text{cos} \theta \text{cos} \rho - \text{sin} \theta \text{sin} \rho \].
This formula helps simplify complex expressions by breaking down the sum of angles into more manageable parts. But why does this matter? In practical applications, such as physics and engineering, this identity is indispensable for solving problems involving wave behavior, oscillations, or even in finding the resultant vector in a plane.

Trigonometric Identities Verification

Verifying trigonometric identities is a fundamental skill in trigonometry. An identity is a mathematical expression that is true for all values of the variables involved. To verify an identity, one typically manipulates one or both sides of the equation to demonstrate that the two sides are equivalent. Basic algebraic skills, knowledge of fundamental trigonometric properties, and the cunning use of certain identities are the tools you need. For instance, to show that \( \text{cos}^2 \theta + \text{sin}^2 \theta = 1 \), one can refer to the Pythagorean identity. Such verifications often require a step-by-step approach, checking at each stage that the manipulations are valid and that they bring you closer to the form you're trying to achieve. Through practice, verifying identities strengthens logical thinking and problem-solving skills, serving students not only in mathematics but in any analytical discipline.

Squaring Trigonometric Functions

When squaring trigonometric functions like sine and cosine, it's essential to understand that you're dealing with the trigonometric functions of an angle raised to the power of two. This operation is notable in several trigonometric identities, including the fundamental \( \text{cos}^2 \theta + \text{sin}^2 \theta = 1 \), known as the Pythagorean identity. Squaring these functions can also lead to other important identities, and it is useful in analysis and calculus when deriving or integrating trigonometric expressions. For complex calculations, remember that while \( \text{cos}^2 \theta \) is just \( \text{cos} \theta \times \text{cos} \theta \), it should not be confused with \( \text{cos}(2\theta) \), the cosine of double the angle, which has its formula and applications.

Cosine Double Angle Identity

The cosine double angle identity is a particular case of the cosine of a sum formula. It gives us valuable information about the cosine of twice an angle. This identity is represented by \( \text{cos}(2\theta) \) and can be expressed in three different but equivalent ways, each highlighting unique properties:

• \( \text{cos}(2\theta) = \text{cos}^2 \theta - \text{sin}^2 \theta \)
• \( \text{cos}(2\theta) = 2\text{cos}^2 \theta - 1 \)
• \( \text{cos}(2\theta) = 1 - 2\text{sin}^2 \theta \)

These forms are instrumental when simplifying trigonometric expressions, solving trigonometric equations, and analyzing waveforms. The choice of which form to use usually depends on the specific problem at hand and which trigonometric function—sine or cosine—is more convenient to work with given what is known or needs to be found.