Problem 73 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\cos ^{2} 5 x-\sin ^{2} 5 x=\cos 10 x$$

Step-by-Step Solution

Answer

The identity is proven using the cosine of double angle formula.

Step 1: Use the Cosine of Double Angle Formula

Start by recalling the double angle formula for cosine: \[\cos 2\theta = \cos^2 \theta - \sin^2 \theta.\]Our task is to show that \( \cos^2 5x - \sin^2 5x \) is equal to \( \cos 10x \). This matches the formula by letting \( \theta = 5x \). Therefore, we replace \( \theta \) by \( 5x \) in the double angle formula.

Step 2: Substitute the Angle

Substitute \( \theta = 5x \) into the double angle identity:\[\cos 2(5x) = \cos^2 (5x) - \sin^2 (5x).\]This becomes:\[\cos 10x = \cos^2 5x - \sin^2 5x.\]

Step 3: Conclusion

We have shown that the expression \( \cos^2 5x - \sin^2 5x \) is equivalent to \( \cos 10x \) as derived from the double angle formula. Therefore, the original identity is proven.

Key Concepts

Cosine Double Angle Formula

The cosine double angle formula is a trigonometric identity used to simplify expressions involving the cosine of twice an angle. This formula is expressed as:\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta. \]This identity is particularly useful in transforming trigonometric expressions, as it directly connects the cosine of a double angle to the squares of sine and cosine of the original angle. Let's break this down:

• The notation \( \theta \) represents the angle you are working with.
• The formula states that the cosine of double that angle can be computed using the difference of the squares of cosine and sine of the angle.

To apply this formula, simply substitute the given angle into \( \theta \) and evaluate both sides of the equation. In our exercise, we replace \( \theta \) with \( 5x \), allowing us to prove the identity \( \cos^2 5x - \sin^2 5x = \cos 10x \). This is a direct application of the cosine double angle formula, showing how useful and versatile it can be for establishing trigonometric identities.

Proof Techniques

Proving trigonometric identities involves a variety of techniques that help demonstrate how one side of an equation can transform into another. One essential approach to proofs in trigonometry is making use of known identities, like the double angle formulas. Here’s why this is helpful:

• These identities provide established shortcuts that simplify complex expressions.
• Utilizing a known formula allows you to apply a straightforward substitution, saving time and effort.

Logical reasoning plays a significant role in proof strategies. This involves:

• Substituting values or expressions directly based on identities.
• Rearranging and grouping terms strategically.
• Verifying the equality of both sides step by step.

When tackling problems like proving \( \cos^2 5x - \sin^2 5x = \cos 10x \), the proof is vastly simplified by recognizing the application of the double angle identity. Always begin by identifying which known identities are applicable, and systematically apply them, ensuring that each step logically leads to the next. This disciplined approach is crucial in developing strong proof skills in trigonometry.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are foundational concepts in mathematics, especially in trigonometry. They describe relationships between the angles and sides of a right triangle, extending into functions that characterize periodic behavior in various applications. Here's a brief overview:

• **Sine (\(\sin\))**: Represents the ratio of the opposite side to the hypotenuse in a right triangle.
• **Cosine (\(\cos\))**: Represents the ratio of the adjacent side to the hypotenuse.
• **Tangent (\(\tan\))**: Typically defined as the ratio of sine to cosine, or the opposite side to the adjacent side.

Understanding these functions is crucial for solving many trigonometric identities and equations. They not only form the building blocks of these identities but also provide tools to model and solve real-world problems involving periodic phenomena like waves and oscillations.In our specific exercise, we applied the cosine function and its related double angle identity to equate two expressions. Recognizing the relationships between these functions is essential for both solving complex equations and proving identities like \( \cos^2 5x - \sin^2 5x = \cos 10x \). Mastery of these fundamental functions provides a solid foundation for advanced mathematical studies and practical applications.