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Chapter 20: Problem 31

prove the given identities. $$\cos ^{2} \alpha-\sin ^{2} \alpha=2 \cos ^{2} \alpha-1$$

Short Answer

Expert verified
The identity is proven: \( \cos^2 \alpha - \sin^2 \alpha = 2 \cos^2 \alpha - 1 \).

Step by step solution

01

Express the Left Side using Pythagorean Identity

We start with the left side of the equation: \( \cos^2 \alpha - \sin^2 \alpha \). Use the Pythagorean identity: \( \sin^2 \alpha = 1 - \cos^2 \alpha \). Substitute this into the equation to get: \( \cos^2 \alpha - (1 - \cos^2 \alpha) \).
02

Simplify the Expression

Simplify the expression obtained from Step 1: \( \cos^2 \alpha - 1 + \cos^2 \alpha \). Combine like terms to get: \( 2\cos^2 \alpha - 1 \).
03

Compare Both Sides of the Equation

Now, compare the simplified expression \( 2\cos^2 \alpha - 1 \) with the right side of the original identity. Both sides are equal, confirming the identity: \( \cos^2 \alpha - \sin^2 \alpha = 2 \cos^2 \alpha - 1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Cosine
Cosine is one of the fundamental trigonometric functions often abbreviated as "cos". It represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. This function plays a crucial role in various trigonometric identities and equations.
  • Range and Domain: The cosine function can take any real number as input. Its output, however, is restricted to the interval between -1 and 1, inclusive.
  • Behavior and Graph: The graph of cosine is a smooth wave-like curve that repeats every 360 degrees, showing its periodic nature. This characteristic is essential in solving many trigonometric problems.
Cosine is crucial when establishing connections with other trigonometric identities, most notably the Pythagorean identity, which we will explore further.
Exploring the Sine Function
Sine, denoted as "sin", is another primary trigonometric function. It is defined as the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle. This function complements cosine in various identities and equations.
  • Range and Domain: Similar to cosine, sine can take any real number as input and outputs values strictly between -1 and 1.
  • Sine Wave: The sine function creates a continuous wave-like pattern, much like cosine, but starts at zero. This pattern is periodic, repeating every 360 degrees or \(2\pi\) radians.
Knowing how sine interacts with cosine is pivotal in transformations and simplifications of trigonometric expressions, enabling us to prove identities such as the ones in our exercise.
The Pythagorean Identity
The Pythagorean identity is a cornerstone in trigonometry, linking sine and cosine in a meaningful equation. The identity can be expressed as \( \sin^2 \alpha + \cos^2 \alpha = 1 \). This indicates that the squares of the sine and cosine of an angle sum to one.
  • Usage: This identity is instrumental in converting between sine and cosine, often simplifying complex trigonometric expressions. It's widely used in proofs, like showing that \( \cos^2 \alpha - \sin^2 \alpha = 2\cos^2 \alpha - 1 \).
  • Flexibility: By rearranging the terms of the Pythagorean identity, either sine or cosine can be isolated. For instance, \( \sin^2 \alpha = 1 - \cos^2 \alpha \), provides a direct means to express sine in terms of cosine, and vice versa.
Through understanding and applying this identity, many seemingly difficult trigonometric problems become more approachable and solvable, as demonstrated in the step-by-step exercise solution.

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