Problem 50 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Use the identity \( \sin^2 x = 1 - \cos^2 x \) and simplify to verify both sides are equal.

Step 1: Recall the Pythagorean Identity

The Pythagorean Identity states that \( \cos^2 x + \sin^2 x = 1 \). This identity is fundamental for verifying trigonometric identities and will be used to simplify the expression.

Step 2: Express \( \sin^2 x \) in terms of \( \cos^2 x \)

Using the Pythagorean Identity, solve for \( \sin^2 x \) by rewriting the identity as \( \sin^2 x = 1 - \cos^2 x \).

Step 3: Substitute the Expression for \( \sin^2 x \)

Replace \( \sin^2 x \) in the original equation \( \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 \) with \( 1 - \cos^2 x \). This gives us: \( \cos^2 x - (1 - \cos^2 x) \).

Step 4: Simplify the Expression

Simplify \( \cos^2 x - (1 - \cos^2 x) \) to get \( \cos^2 x - 1 + \cos^2 x = 2\cos^2 x - 1 \). The terms combine to show both sides are equal, verifying that the identity holds.

Key Concepts

Pythagorean identity

The Pythagorean Identity is a cornerstone in trigonometry, drawing a direct link between the square of the sine and cosine functions. It is expressed as \( \cos^2 x + \sin^2 x = 1 \). This relation is similar to the Pythagorean Theorem in geometry, where the sum of the squares of the legs of a right triangle equals the square of the hypotenuse. This identity helps in transforming and verifying trigonometric expressions.
When tackling problems or verifying identities, we often isolate one term using this identity:

• \( \sin^2 x = 1 - \cos^2 x \)
• \( \cos^2 x = 1 - \sin^2 x \)

Utilizing this identity allows us to replace and simplify complex expressions, which is crucial in proving further trigonometric identities and solving trigonometry problems.

cosine function

The cosine function, denoted as \( \cos(x) \), represents the ratio between the adjacent side and the hypotenuse in a right triangle. It is one of the primary trigonometric functions and is periodic with a period of \( 2\pi \). This means it repeats its values every \( 2\pi \) radians.
Characteristics of the cosine function include:

• Range: \([-1, 1]\)
• Even function: \( \cos(-x) = \cos(x) \)
• Graph: A wave-like curve starting from 1 when \( x = 0 \)

In the given identity, \( \cos^2 x \), we see the cosine function being squared. The square of the cosine maintains non-negativity of values and becomes crucial when manipulating expressions involving both \( \cos \) and \( \sin \). Understanding these traits is essential for solving trigonometric problems, especially when proving identities or calculating angles.

sine function

The sine function, \( \sin(x) \), is the ratio of the opposite side to the hypotenuse in a right triangle. Like cosine, sine is a primary trigonometric function defined for all angles. It also has a periodic nature, repeating its values every \( 2\pi \) radians.
Key features of the sine function include:

• Range: \([-1, 1]\)
• Odd function: \( \sin(-x) = -\sin(x) \)
• Graph: Starts from 0 at \( x = 0 \) and follows a smooth wave

In the example problem, \( \sin^2 x \) is used, derived from the Pythagorean Identity. It illustrates that \( \sin \) squared will blend with the cosine squared terms to simplify expressions. Understanding sine's behavior is vital when composing or deconstructing trigonometric identities, as it's frequently interlinked with \( \cos \) through identities.