Problem 50 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1$$

Step-by-Step Solution

Answer

The identity is verified using the cosine double angle identity.

Step 1: Identify the Trigonometric Identity

We are given the equation \( \cos^{2} x - \sin^{2} x = 2 \cos^{2} x - 1 \). Our goal is to verify this identity. First, recognize that this is a form of the double angle identity for cosine. The double angle identity for cosine is \( \cos(2x) = \cos^{2} x - \sin^{2} x \) and can also be written as \( \cos(2x) = 2 \cos^{2} x - 1 \).

Step 2: Match the Given Equation to Known Identity

Notice that both sides of the given equation correspond to different forms of the cosine double angle identity: \( \cos^{2} x - \sin^{2} x \) is another expression for \( \cos(2x) \), as is \( 2 \cos^{2} x - 1 \). Therefore, the given equation is indeed an identity, as both sides of the equation represent \( \cos(2x) \).

Step 3: Conclusion

Since both sides of the equation \( \cos^{2} x - \sin^{2} x = 2 \cos^{2} x - 1 \) can be rewritten as \( \cos(2x) \), this demonstrates that the two expressions are equivalent and thus the identity is verified.

Key Concepts

Double Angle Identity

The double angle identities are a set of trigonometric identities that allow us to express trigonometric functions of double angles, like \(2x\), in terms of single angles, such as \(x\). These identities are particularly useful because they provide a way to simplify expressions and solve problems involving trigonometric equations. One of the most commonly used double angle identities is the one for the cosine function:\[ \cos(2x) = \cos^2(x) - \sin^2(x) \]This identity is derived from the Pythagorean identity and expresses \(\cos(2x)\) in a form that involves both cosine and sine. Another version of the same identity makes it suitable for manipulating expressions involving only cosine: \[ \cos(2x) = 2 \cos^2(x) - 1 \]These identities are extremely useful when verifying trigonometric identities, simplifying expressions, and solving trigonometric equations.
It's important to be familiar with these formulas and understand how to manipulate them to fit different forms of problems.

Cosine Function

The cosine function is one of the primary trigonometric functions. It represents the x-coordinate of a point on the unit circle as an angle \(x\) is swept around. The function oscillates between -1 and 1, creating a wave-like pattern over its cycle. In the realm of identities, cosine plays a crucial role. Knowing the behavior of the cosine function can help in applications such as verifying identities, modeling periodic phenomena, and solving triangles. Properties of the cosine function include:

• Its period is \(2\pi\), meaning that \(\cos(x) = \cos(x + 2\pi k)\) for any integer \(k\).
• It is an even function, which means that \(\cos(-x) = \cos(x)\).
• It starts at 1 when \(x = 0\), decreases to -1 at \(\pi\), then returns to 1 at \(2\pi\).

These properties make the cosine function especially suitable for expressing relationships between angles and verifying identities. Cosine is an integral part of the double angle identities as well, serving as both the subject and component of these formulas.

Verifying Identities

Verifying trigonometric identities involves showing that two trigonometric expressions are equivalent. This requires using known identities and algebraic manipulations to transform one side of the equation into the other.In our example of verifying \(\cos^2(x) - \sin^2(x) = 2 \cos^2(x) - 1\), we used a specific double angle identity to prove the expressions equal. This process generally involves:

• Identifying familiar patterns or known identities within the expressions.
• Applying algebraic transformations like factoring or expanding.
• Simplifying each side of the equation to a common form, often reducing them to a basic trigonometric function such as \(\cos(2x)\).

The key is patience and a solid understanding of trigonometric formulas.
Practice helps improve the ability to quickly recognize and utilize the appropriate identities, which can simplify otherwise complex relationships in trigonometry.