Problem 59 - Algebra and Trigonometry

Question

59–76 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 double angle formula for cosine.

Step 1: Recall the Double Angle Formula for Cosine

The identity we want to prove can be expressed in terms of a known trigonometric identity. We recall the double angle formula for cosine, which states: \[ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \] This is the core identity we'll use to prove the given expression.

Step 2: Identify the angles and compare

In our expression, we have \(\cos^2(5x) - \sin^2(5x)\). Notice that this matches the structure of the double angle formula if we let \(\theta = 5x\), then \[ \cos(2\theta) = \cos(10x) \] This means that \(\cos^2(5x) - \sin^2(5x) = \cos(10x)\) directly follows from the double angle formula for cosine by substituting \(\theta\) with \(5x\).

Step 3: Conclude the Identity Proof

Since we determined that the expression \(\cos^2(5x) - \sin^2(5x)\) indeed matches the double angle formula expression \(\cos(10x)\), the given identity is confirmed correct. Thus, we have shown that:\[ \cos^2(5x) - \sin^2(5x) = \cos(10x) \] This completes our proof.

Key Concepts

Double Angle Formulas

Double angle formulas are fundamental tools in trigonometry that help simplify expressions involving trigonometric functions. They provide a way to express trigonometric functions of double angles in terms of single angles. Particularly, the double angle formula for cosine is expressed as: \[\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\] This formula reveals that the cosine of a double angle can be rewritten using the squares of the sine and cosine functions of the original angle.
This identity is extremely helpful for proving other trigonometric identities or simplifying trigonometric expressions. For instance, to prove that \( \cos^2(5x) - \sin^2(5x) = \cos(10x) \), you can substitute \(\theta = 5x\) in the double angle formula. By recognizing patterns in trigonometric expressions, you can often reduce complex problems to simpler known identities using these formulas. Understanding and applying double angle formulas is a key skill in trigonometry.

Cosine Function

The cosine function is one of the basic trigonometric functions related to the ratio of the adjacent side to the hypotenuse in a right triangle. This periodic function is fundamental and has important properties:

• It is an even function: \( \cos(-\theta) = \cos(\theta) \)
• Its range is from -1 to 1.
• It repeats every \(2\pi\) radians, indicating its periodicity.

In the context of the double angle formula, the cosine function helps derive expressions like \( \cos(2\theta) \) which simplifies using \( \cos^2(\theta) - \sin^2(\theta) \).
This versatility allows it to be used in various proofs and trigonometric manipulations. The key is recognizing how the cosine function operates in the identity and how it can transform one trigonometric expression into another, sometimes more simplified form. Mastery of the cosine function and its properties is crucial for tackling trigonometry problems with ease.

Proofs in Trigonometry

Proofs in trigonometry are logical arguments used to verify the truth of trigonometric identities and equations. They often involve transforming one side of an equation to match the other using known identities and algebraic manipulations.
To effectively conduct proofs, you need a solid understanding of key trigonometric identities like the Pythagorean identities, double angle formulas, and angle addition formulas. In the example of proving \( \cos^2(5x) - \sin^2(5x) = \cos(10x) \), you leverage the double angle formula for cosine to transform the left-hand side to match the right-hand side.

• Recognize the identity: Utilize known identities to transform expressions.
• Apply algebraic manipulations: Move terms and combine like terms as necessary to simplify expressions.
• Verify equality: Ensure both sides of the identity are equal after transformation.

Proofs in trigonometry not only establish trust in mathematical relationships but also enhance problem-solving skills by encouraging deep understanding of trigonometric concepts and their interconnections. Practicing these methods allows students to gain confidence and capability in handling trigonometric identities and proofs.